The Odyssey is an ancient Greek epic poem following the wanderings of Odysseus on his return from the Trojan War. As such, it is a sequel to the Iliad, although its exact relationship to the earlier poem is as controversial as the historical existence of Homer, the traditional author of both epics. As with the Iliad, the story opens 'in the middle of things' with Odysseus held captive by the goddess Calypso. His adventures in the ten years since the fall of Troy are recounted as the story advances towards his final homecoming.
Free online texts
Gutenberg: The Odyssey, translated by Alexander Pope. Multiple formats.
Gutenberg: The Odyssey, done into English prose, translated by Samuel Henry Butcher and Andrew Lang. Multiple formats.
Gutenberg: The Odyssey, Rendered into English prose for the use of those who cannot read the original, translated by Samuel Butler. Multiple formats.
Poetry in Translation: The Odyssey, translated by A.S. Kline (2004). Multiple formats.
Theoi.com: The Odyssey. Translated by Murray, A T. Loeb Classical Library Volumes. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1919.
University of Adelaide (Internet Archive): The Odyssey, translated by by Samuel Butler. EPUB, HTML and MOBI formats.
Bartleby: The Adventures of Odysseus and the Tale of Troy, retold by Padraic Colum (Macmillan, 1918). HTML text.
The Conversation: Guide to the Classics - Homer's Odyssey, by Professor Chris Mackie.
Greek Myth Comix: Odyssey Comix, by Laura Jenkinson.
Gutenberg: Homer's Odyssey: A Commentary, by Jacques Denton Snider.
In Our Time: The Odyssey - BBC radio discussion with Melvyn Bragg.
Librivox: The Odyssey - public domain audiobooks.
TedEd: The science behind the myth: Homer's "Odyssey" - video by Matt Kaplan
Vox: Historically, men translated the Odyssey. Here’s what happened when a woman took the job, by Anna North, 20 Nov 2017.
Youtube: A Long and Difficult Journey, or The Odyssey - video from the CrashCourse channel.
Further reading at Tom's Learning Notes
Ancient Greek resources: Learn to read Greek classics in the original.
Homer: The Iliad.
Euripides: Cyclops - dramatises one of the most famous episodes from the Odyssey.
Bloom's Western Canon - The Odyssey is listed.