Odysseus encountered the monstrous Cyclops named Polyphemus on his journey home in The Odyssey. This one-eyed giant is known for his brute strength and fierce temperament. In the story, Odysseus and his men become trapped in Polyphemus's cave, leading to a cunning escape plan orchestrated by Odysseus. The encounter highlights themes of intellect over strength and the consequences of human folly, exemplifying Odysseus's role as a crafty hero in Greek mythology.