Based on the "Odyssey" by Homer
The titles acknowledge the work is, "Based upon Homer's Odyssey." In interviews, the Coen brothers say that they never actually read the Odyssey, but the film patterns many of its characters and situations upon that ancient work (see below). The Coens claim to having gleaned most of the incidental details of Homer's epic from the 1955 filmed adaptation, starring Kirk Douglas.
Episodes in the film somewhat resemble the Odyssey. Everett tends to come off much worse than his mythical counterpart Odysseus. A viewer familiar with the Odyssey may often expect the hero to triumph. Typically, this does not happen, although things turn out all right later, partly because the hero is so irrepressible, partly by sheer luck. Among the similarities:
"Sing to me of the man, Muse... ", the line at the beginning of the film, is the first line of the Odyssey.
Ulysses is the Roman name of Odysseus.
A blind prophet - possibly a Tiresias figure - who foreshadows that "the treasure you seek shall not be the treasure you find."
A merciless sheriff wanting to lynch him, perhaps analogous to Poseidon in the story of the Odyssey, but also alluded to as Satan throughout the film; this is strengthened by the presence of a hunting dog, which echoes Cerberus as well as the common mythological Hellhounds. The link between the two (Satan and Poseidon) is made when Ulysses mentions that Satan carries "a giant fork" (a trident); both figures are often depicted with just such an instrument.
http://en.wikipedia.org/wiki/O_Brother,_Where_Art_Thou%3F