O Brother Where Art Thou? has several allusions to The Odyssey in its narrative, themes, and characters.
Firstly, the George Clooney character, Ulysses, is based directly on Odysseus. (Ulysses is the Latin variation of the name Odysseus, which is important to keep in mind.) Like Odysseus, he yearns to return home and reunite with his wife (called Penny instead of Penelope in this story). He undergoes a journey wherein he meets strange and menacing figures who want to prevent him from achieving his goal for a variety of reasons.
Several characters in the film reference corresponding figures in Homer. The blind prophet Tiresias is replaced with an old blind man on the railway who also tells future events to the protagonist and his companions. The sirens are replaced with three shapely wash-women who drug the men with liquor. The KKK become a stand-in for the Cyclops, with the main characters disguising themselves as clansmen the same way Odysseus and his men used sheep's wool to disguise themselves during their escape.
And so on it goes. Though the plot isn't exactly the same as The Odyssey, the movie riffs a lot on the epic poem.