If I were preparing a homework assignment on Woody Allen, it would focus on the influences on his comedy. Allen, born Allen Konigsberg, has written plays, books, screenplays, and articles for magazines (mainly, The New Yorker). In short, there is a lot of material from which to draw.
Ironically, one of the most important influences on Allen's career was the Swedish film director Ingmar Bergman, whose films could hardly be categorized as comedies. On the contrary, Bergman's films are solemn, reflective, and dark. Allen's homage to the Bergman influence has been highly visible. Allen's comedy about Russia under the czars, "Love and Death," draws heavily from Bergman's work. His one serious, and critically and commercially unsuccessful, film, "Interiors," was the most reflective of Bergman's influence.
In addition to Bergman, there have many comedians to whom Allen has paid homage. Allen got his start in television as a young comedy writer for the early-1950s television programs headlined by Sid Caesar. Allen, alongside other future comedy greats like Mel Brooks, Carl Reiner and Neil Simon, got their start writing for Caesar, some on the latter's "Your Show of Shows," and others, including Allen and later MASH producer Larry Gelbart, on other Caesar projects. It is almost certain that at least some of Allen's material is a product of his collaboration with these great comedic minds.
A constant theme of Allen's work is, of course, his neuroses, including his fascination with death, religion and philosophy. Allen's Jewish upbringing is interwoven through these themes, for example, references to anti-Semitism throughout his work. The Allen character's paranoia is legendary, as, in "Annie Hall," when he insists that another person, in response to the question "did you eat?" replies, "No, Jew." The Allen character's paranoia, combined with fear of anti-Semitism, has him convinced that the answer "No, did you?" was actually "No, Jew." Similarly, in the same film, Allen's character is convinced that Annie's grandmother is a rabid anti-semite.
All of these themes were present in Allen's stand-up comedy routines of the 1960s, for example, his reference to his obviously Jewish grandmother having been raped by Cossaks when the family lived in Russia -- a joke meant to remind listerners of his Jewish heritage and the victimization to which Jews were subjected.