James Joyce's novel Ulysses is based on Homer's Odyssey, and many of the references in Ulysses are drawn from it. The first obvious allusion is the title. Ulysses is the Latin name for Odysseus. The novel itself is set in a specific time and place, the day of 16 June 1904 in the city of Dublin, and was first published in serial form in 1918-1920 and in book form in 1922. In this period, students began studying Latin quite early in their school careers, and then began Greek somewhat later, in secondary school. Traditional language pedagogy often involved translating Greek into Latin, and thus what we have in the novel is a layering of an Irish version of a Latin-inflected prose adaptation of a Greek epic.
The first allusion we encounter in the book is the name Stephen Dedalus. It is based on the ancient myth of Daedalus, a Greek inventor, who with his son Icarus was imprisoned by King Minos. Daedalus fashioned wings of bird feathers and wax and he and Icarus escaped the tower in which they were imprisoned by flying with the mechanical wings, but Icarus, despite his father's warning flew to close to the sun, melting the wax holding his wings together, and died.
The structure of Ulysses mimics that of the Odyssey, with specific sections of the novel given titles highlighting the parallels. The section introducing Stephen is the Telemachia, a term used to describe the section in the Odyssey detailing the adventures of Odysseus' son, Telemachus. The main section, the experiences of Leopold Bloom, parallel the adventures of Odysseus, and the passages concerning Molly Bloom parallel the sections of the Odyssey devoted to Odysseus' wife Penelope.
The various obstacles and temptations Leopold surmounts on his way home also allude to similar elements in Homer, with the lotus eater section talking about drugs, Hades about a funeral, and the Sirens being two barmaids.
See the full eNotes summary for additional examples of allusions.