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What is the oldest European royal or princely house, either enthroned or dethroned, that patrilineally survives nowadays?

In terms of continuously dateable genealogy, it is probably the Bagratids of Georgia, the current head of which is disputed between three branches.
The Georgian branch was founded by Adarnase in the late 700s as branch of the Armenian Bagratuni dynasty, though descendants then fabricated an origin story claiming descent from the biblical David, which is obviously fanciful fiction. The Bragatids may have had recorded ancestors as far back as classical antiquity, but the intervening genealogy is basically non-existent.

Another candidate is the House of France, founded in 987 by Hugh Capet. It current survives in the legitimate male line through a cadet branch, headed by Jean d'Orléans. Hugh Capet was himself a male line descendant of the Robertians, and thus the Capetians can trace their descent at least as far back as Robert the Strong, and probably to Robert II of Worms, before 800. The earliest ancestors of the Robertian lineage can be pushed as far back as Charibert of Hesbaye in the 500s, but very little is known about him and the intervening genealogy is not definitive.

Realistically, there's probably any number of ancient dynasties that survive to this day, but without genealogical records to identify their descendants. This is particularly true because the question did not specify legitimate issue - unfaithful princes are legion in history, and with enough bastards you can win the probability game. For the most part however, bastards and non-inheriting sons (who do not otherwise distinguish themselves), as well as dispossessed houses in general, rapidly fade out of history.

The House of Plantagenet, for instance, is usually held to be extinct in the legitimate male line with the execution of Edward in 1499. However, it actually survives in the male line through Henry Somerset, whose direct male line ancestor John Beaufort was a legitimised bastard of John of Gaunt. Note that the first Plantagenet, Geoffrey V of Anjou inherited his title cognatically, but was an agnatic descendant of the less known Geoffrey I of Châteaudun. This Geoffrey in turn seems to have been a direct agnatic descendant Rorgon I of Maine. This would mean that the Rorgonid dynasty, established before 819, survives today through Henry Somerset.

These examples illustrate a second problem, which is that the earliest genealogy of older dynasties are actually very unclear. It is thus very difficult to date the age of a dynasty accurately. Although there are conventional founding dates, the founders were themselves members of powerful established dynasties, so it is essentially a judgement call to determine where a dynasty begins.

While outside Europe, an honourable mention goes to the House of Confucius. His 79th generation descendant in the direct agnatic line, Kung Tsui-chang, still holds title as the hereditary Sacrificial Official in Taiwan. This is a ministerial level post created as a Republican continuation of the Holy Dukes of Yen, after the Chinese Revolution abolished every other title of nobility.

Confucius was himself a 16th generation descendant in the male line of Di Yi, who was the penultimate king of Shang China. That royal dynasty was founded by Chen Tang when he was said to have overthrown the previous (but legendary) Xia dynasty circa 1675 BC.

Thus, although his line hasn't been sovereign for some 3000 years, Kung Tsui-chang is an agnatic descendant of the oldest archaeologically-confirmed ruling house of China.

Note on the terminology:

This question seems to have attracted some confusion over what "dynasty" means. While dynasty is not generally a strictly well defined term, genealogical descent is. A lineage that is traced exclusively through male offsprings is patrilineal, or agnatic. Those that goes through any combination of male or female links is called cognatic. A dynasty can thus be simultaneously extinct in the agnatic line while persevering in the cognatic line.

For example, while Queen Elizabeth is technically the last agnatic member of the British Saxe-Coburg and Gotha line to reign, the Windsor dynasty continues cognatically though her children, e.g. Prince Charles. Likewise, the House of Austria was technically extinct with the death of Maria Theresa in 1780, but survives cognatically through her children as the House of Habsburg-Lorraine. An even earlier example is the House of Welf, which became extinct in the male line with the death of Welf III in 1055, but survives in the cognatic line through his sister to this very day.

Since the question specifies patrilineal descent, cognatic survivals are excluded.

"The oldest noble family" is a somewhat fictional concept: When a 'House' starts or ends is somewhat arbitrary, and not uniformly handled throughout European history. There were countless exceptions, uncertainties etc. Further complicated by date of ennoblement, as none of them fall from the skies or heavens, and achieved level of nobility, the following are all viable contenders (to be read in addition to semaphore's list):

Reginarids or old, but only started with documented certainty with Gilbert, Count of the Maasgau (mentioned in 841) and survive today in House of Hesse.

House of Welf, from Hundings in legend and Welf I, Duke of Bavaria in reality (*~1030)) currently a darling of the yellow press Prince Ernst August of Hanover. The branch Welf-Este:

According to Edward Gibbon, the family originated from the Roman Attii family, which migrated from Rome to Este to defend Italy against the Ostrogoths.

Otto I, Count of Scheyern (*~1020) founder of the House of Wittelsbach has a descendent in Franz, Duke of Bavaria

House of Ascania, founded 1036 survives today in Eduard, Prince of Anhalt.

House of Wettin, founded 900s, survives today in Michael, Prince of Saxe-Weimar-Eisenach.

As the last name might indicate, the Wettins are the parent branch of currently ruling members of Saxe-Coburg-Gotha, namely in England and Belgium.

House Nassau was founded in 1093 and was dissolved according to its German origin tradition only in 1985, after ruling in the Netherlands for while. However, this agnatic-only rule was abolished like in England, and so the current head is Henri, Grand Duke of Luxembourg.

But that is of course one of the absurdities enshrined into the concept of counting purely agnatic descent, as only mater semper certa est!

To emphasise the arbitrariness of these rules we might look first at the Windsors (Saxe-Coburg-Gotha) and how through all these intermarriages with other noble families they replaced Hannoverian rule in England.

Similarly the Habsburgs were practically extinct according to the rules, yet the name and rule continued:

The House of Habsburg became extinct in the 18th century. The senior Spanish branch ended upon the death of Charles II of Spain in 1700 and was replaced by the House of Bourbon. The remaining Austrian branch became extinct in the male line in 1740 with the death of Holy Roman Emperor Charles VI, and completely in 1780 with the death of his eldest daughter Maria Theresa of Austria. It was succeeded by the Vaudémont branch of the House of Lorraine, descendants of Maria Theresa's marriage to Francis III, Duke of Lorraine. The new successor house styled itself formally as the House of Habsburg-Lorraine (German: Habsburg-Lothringen), and because it was often confusingly still referred to as the House of Habsburg, historians use the unofficial appellation of the Habsburg Monarchy for the countries and provinces that were ruled by the junior Austrian branch of the House of Habsburg between 1521 and 1780 and then by the successor branch of Habsburg-Lorraine until 1918. The Lorraine branch continues to exist to this day and its members use the Habsburg name (example: Otto von Habsburg).

Even greater in its fanciness are the Salms, known from well before 1000 and even providing a German anti-king during the Great Saxon Revolt in 1081.

The descendants of Elizabeth and Frederick became extinct in 1416. Their possessions were inherited by the Lords of Reifferscheid, who resided at Reifferscheid Castle. The succession arrangement was challenged by the Raugraves, however, they had to accept a 1456 judgement by the Luxembourg councillor Antoine I de Croÿ.

House, or family name is a form of title and possession in this case.
Meaning technically all lines of this family went extinct, multiple times (!), yet they have surviving family members, bearing the name.

Also a note on definitions:

Genalogical descent isn't that well – or strictly – defined here either; or better: traceable with certainty. Not to warm up an old debate (or 'theory'), but as an example, even paternity for Harry was disputed. Add to that all the exceptions to agnatic inheritance and Salic law, now even the complete abolishment of said 'rule' in almost all families, it should become clear that fans of nobility cling to a fictional concept, distinct from strict genealogy and subject to changing laws and conventions.

As Steve Bird already commented below the question:

Given the high degree of intermarriage between the European royal houses, it's not all that easy to tell them apart...so all of them? ´

Except one.

The only House still claiming to practicing this form of 'law' is the House of Liechtenstein.

To reiterate: Harry, just as an example, may be a patrilineal descendent of House Oldenburg, but, by act of law, he is now not:

Harry is a male line descendant of Elimar I, Count of Oldenburg, and a member of the House of Oldenburg, one of Europe's oldest royal houses; the cadet branch to which he belongs, known as the House of Glücksburg, was founded by his paternal ancestor Friedrich Wilhelm, Duke of Schleswig-Holstein-Sonderburg-Glücksburg. Harry's paternal grandmother, Queen Elizabeth II, issued letters patent on 8 February 1960 declaring his father to be a member of the House of Windsor. His male line ancestors include eleven Counts of Oldenburg, two dukes of Schleswig-Holstein-Sonderburg, five dukes of Schleswig-Holstein-Sonderburg-Beck, a duke of Schleswig-Holstein-Sonderburg-Glücksburg, four Danish kings – Christian I, Frederick I, Christian III, Christian IX – and King George I of Greece.

A similar pattern then arises with from 1131–1918 so long reigning Obodrites. Which are said to survive until today in male line. But:

With the extinction of Schwerin, Mecklenburg-Strelitz is now the only surviving branch of the Grand Ducal house in the male line. The current head of this house is Borwin, Duke of Mecklenburg. His grandfather was Count Georg of Carlow, the morganatic son of Duke George Alexander of Mecklenburg (1859–1909). Georg was adopted in 1928 by his uncle Duke Charles Michael of Mecklenburg, the head of the House of Mecklenburg-Strelitz. He then assumed the title and style of "His Serene Highness The Duke of Mecklenburg", which was confirmed by the head of the Imperial House of Russia, Grand Duke Cyril Vladimirovich on 18 July 1929 and recognised on 23 December by Grand Duke Friedrich Franz IV of Mecklenburg-Schwerin.4 He succeeded his uncle as head of the house on 6 December 19345 and was granted the style of Highness on 18 December 1950.4

As historians we can go back a few generations and identify people as then being 'a member of a house of X'. As long as we have records, we can also say with certainty, 'who was the mother of Y'. But combining the nobility concept of "house of" with patrilinearity and going back to the middle ages to construct any "survives until today" narrative makes us enter fictional territory with a an unsound basis in conventions and law.

Compare these tales with the story of the Bagratides from Georgia. Who is their anscestor? If we ask them, they will deny any connection to a family of the very same name from Armenia, they say it's a man called David. Pretty well known

Short Answer: Nobody knows for sure.

Long Answer It is very complicated.

A royal pedigree in the agnatic (male only) line that is more or less totally proven and accepted by everyone who studied the subject is likely to be shorter by many generations, centuries, and possibly even millennia, than the longest pedigree ever used by that same royal family but which is doubted and denied by most or all historians.

Many of the royal dynasties of Europe sometimes used pedigrees tracing their ancestry back to Christian and Jewish legendary and mythical persons, back to Noah and thus to Adam, ancestor of all humans according to myth. Sometimes those pedigrees went back to the royal family of Troy and to Norse and/or Greek gods, believed in the middle ages to have been men who became worshiped as gods.

And in real life, according to the scientific viewpoint, no matter how long or how short the recorded agnatic pedigree of any person is, his actual agnatic pedigree is many times longer. Each and every man who ever lived had a father, and each father had a father, and so on and so on, back generation after generation of men, each with a father, back to the hypothetical Y-chromosomal most recent common ancestor or Y-chromosomal Adam, who supposedly lived about 200,000 to 300,000 years ago, and on beyond him, for generation after generation of prehuman beings, back tens of millions and hundreds of millions of years to the first multi celled animals, and back farther and farther to the beginning of sexual reproduction.

So in one sense all royal dynasties and every other family go back the same amount of time, go back the same number of millions of years. But in another sense each family or royal dynasty goes back only as far back as its agnatic pedigree is recorded reliably. And of course there is often considerable doubt and controversy about how far back the agnatic pedigree of a particular dynasty is recorded reliably. Thus there can easily be centuries of uncertainty in the age of a specific dynasty.

For example, Semaphore asked which member of the Merovingian dynasty is alive today. And the correct answer is that nobody today has a proven descent from the Merovingian dynasty, let alone a proven agnatic descent. But that didn't stop some writers from creating pedigrees making the Carolingian and Capetian dynasties descended from younger branches of the Merovingian dynasty, or stop Emperor Maximilian I from claiming the Habsburgs were agnatic descendants of the Merovingians.

My answer to this question:

https://worldbuilding.stackexchange.com/questions/124553/how-do-i-make-my-dynasty-last-1000s-of-years/124738#1247381

Includes a list of the ten longest lasting dynasties in history (from Europe, Asia & Oceania), which I criticize for accepting too many implausible and unsupported lengths of dynasties.

I then follow it with my own list of the 20 longest lasting dynasties from Europe, Asia, Africa, and Oceania I could think of at that time. Note that I usually list several different possible periods that those dynasties might have ruled or reigned for, due to various uncertainties.

Here is a link to a thread "What Were the Longest Lasting Dynasties in Europe?":

https://historum.com/threads/what-were-the-longest-lasting-dynasties-in-europe.139428/2

Several persons suggested a number of candidates, and I replied by pointing out that the proven antiquity of most of those dynasties was a lot shorter than the claimed antiquity.

See my post number 12 on page 2. The Capetian dynasty is the oldest of the ones listed in post number 12. And in post number 20 on page 2 I discuss whether the Bagrationi dynasty of Georgia, suggested as a possibility by Semaphore, was one, two, or three agnatic dynasties.

In post number 16 on page 2 I list the lengths of historical existence of 39 Irish kingdoms based on the reigns of their kings listed in the Regnal Chronologies website.

http://my.raex.com/~obsidian/irel.html#Ardry3

According to that data some of those kingdoms lasted for very long times, six of them for over a thousand years.

And here is a link to a list of "Gaelic nobility of Ireland" listing various families descended from medieval Irish kings.

https://en.wikipedia.org/wiki/Gaelic_nobility_of_Ireland4

And the length of that list shows that there are a number of Irish families tracing their ancestry back for in many cases over a thousand years and up to maybe 1,500 years before reaching the legendary period of their ancestry.

On the other side of Europe, there are a number of possibly quite old Greek families, the Phanariotes.

Phanariotes, Phanariots, or Phanariote Greeks (Greek: Φαναριώτες, Romanian: Fanarioți, Turkish: Fenerliler) were members of prominent Greek families in Phanar2 (Φανάρι, modern Fener),3 the chief Greek quarter of Constantinople where the Ecumenical Patriarchate is located, who traditionally occupied four important positions in the Ottoman Empire: Grand Dragoman, Grand Dragoman of the Fleet, Hospodar of Moldavia, and Hospodar of Wallachia. Despite their cosmopolitanism and often-Western education, the Phanariotes were aware of their Hellenism; according to Nicholas Mavrocordatos' Philotheou Parerga, "We are a race completely Hellenic".4

https://en.wikipedia.org/wiki/Phanariotes5

Many of the Phanariotes families listed in the article might be extinct in the agnatic (male only) line, but it is probable that many still continue.

Many of the Phanariotes families have the same surnames as prominent noble families of the eastern Roman or "Byzantine" empire. Some have the names of former imperial dynasties:

Angelos, Kantakouzenos or Cantacuzenus, Doukas or Dukas, Comnenus or Komnenos, and Palaiologos or Palaeologus.

But in the "Byzantine" era it was common for men to take the surnames of their maternal ancestors or even of unrelated families, especially if they were prominent families. Thus it is uncertain what percentage of the Phanariotes families are descended in the male line from famous "Byzantine" families, as well as which Pharariotes families are so descended.

The family that is considered most likely to actually be descended from their alleged "Byzantine" ancestors is the Cantacuzenos family which claims to be descended from Emperor John VI Kantakouzenos (c. 1292-1383) and his son and co emperor Matthew Kantakouzenos (c. 1325-1383). The first known Kanakouzenos was an officer in the reign of Alexios I Komnenos (reigned 1081-1118).

The Russian princes Andronikov claimed to be descended from the Andronikashvili Princes in Kakheti in Georgia who claimed descent from the evil Emperor Andronikos I Komnenos.

https://en.wikipedia.org/wiki/Andronikashvili6

If that claim, or the claim of the Phanariotes Komnenos family, is correct, they would trace their ancestry back to Manuel Erotikos Komnenos (c.955/60-c. 1020).

And back in Britain there are a few Welsh royal families tracing their ancestry back as far as Irish royal families.

The Anwyl of Tywyn family traces their agnatic ancestry to King Owain Gwynedd of Gwynedd and the House of Aberffraw.

https://en.wikipedia.org/wiki/Anwyl_of_Tywyn_family7

The House of Aberffraw goes back to King Merfyn the Freckled who started to reign about 825 and whose ancestry was traced back to King Coel Hen ("the old") about AD 400 and back to Beli Mawr about 100 BC and back to the Trojan royal family and to Adam.

The Williams family of Aberpergwm is descended from Jenkin William who settled at Aberpergwm about 1500, who was a descendant of Morgan Gam (d. 1241), a descendant of Iestyn ap Gwrgant (fl, c. 1081-1095), the last king of Glamorgan or Morganweg, who was a great great grandson of King Morgan the Old who might have died about 974, who was descended from centuries of kings of the region.

And no doubt someone could write a entire book trying to figure out which European royal family is the oldest.