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I read in another question that Fox produced the film I, Robot because they had just acquired the rights to the Asimov collection and wanted to get a big budget movie out of it. I would have thought his books are old enough to already be in the public domain, but obviously I'm not very familiar with the law. When does the copyright expire on his various works?

This question is covered by the Asimov FAQ:

All of Asimov's work, fiction and non-fiction, was under copyright at the time of the Good Doctor's death. Under current U.S. law, the copyrights for his works published before 1978 will not expire until 95 years after the copyrights were obtained, and those published from 1978 onward will remain in effect for 70 years after his death. Thus these copyrights will remain in effect until dates ranging from 2034 (for his first story published in 1939), through 2072 (for works published in 1977), and works published from 1978 onward will remain under copyright until 2062, although the relevant laws may change between now and then.

(emphasis is mine)