Polyphemus (/ˌpɒliˈfiːməs/ ^ⓘ; Ancient Greek: Πολύφημος, romanized: Polyphēmos, Epic Greek: [polypʰɛːmos]; Latin: Polyphēmus [pɔlʏˈpʰeːmʊs]) is the one-eyed giant son of Poseidon and Thoosa in Greek mythology, one of the Cyclopes described in Homer's Odyssey. His name means "abounding in songs and legends", "many-voiced" or "very famous". Polyphemus first appeared as a savage man-eating giant in the ninth book of the Odyssey. The satyr play Cyclops by Euripides is dependent on this episode apart from one detail: Polyphemus is made a pederast in the play. Later Classical writers presented him in their poems as heterosexual and linked his name with the nymph Galatea. Often he was portrayed as unsuccessful in these, and as unaware of his disproportionate size and musical failings. In the work of even later authors, however, he is presented as both a successful lover and skilled musician. From the Renaissance on, art and literature reflect all of these interpretations of the giant.

In Homer's epic, Odysseus lands on the island of the Cyclopes during his journey home from the Trojan War and, together with some of his men, enters a cave filled with provisions. When the giant Polyphemus returns home with his flocks, he blocks the entrance with a great stone and, scorning the usual custom of hospitality, eats two of the men. Next morning, the giant kills and eats two more and leaves the cave to graze his sheep.

After the giant returns in the evening and eats two more of the men, Odysseus offers Polyphemus some strong and undiluted wine given to him earlier on his journey. Drunk and unwary, the giant asks Odysseus his name, promising him a guest-gift if he answers. Odysseus tells him "Οὖτις", which means "nobody" and Polyphemus promises to eat this "Nobody" last of all. With that, he falls into a drunken sleep. Odysseus takes a wooden stake he had earlier hardened in the fire, returns it to the fire until it is glowing hot, and drives it into Polyphemus's eye. When Polyphemus shouts for help from his fellow giants, saying that "Nobody" has hurt him, they think Polyphemus is being afflicted by divine power and recommend prayer as the answer.

In the morning, the blind Cyclops lets the sheep out to graze, feeling their backs to ensure that the men are not escaping. However, Odysseus and his men have tied themselves to the undersides of the animals and so get away. As he sails off with his men, Odysseus boastfully reveals his real name, an act of hubris that was to cause problems for him later. Polyphemus prays to his father, Poseidon, for revenge and casts huge rocks towards the ship, which Odysseus barely escapes.

The story reappears in later Classical literature. In Cyclops, the 5th-century BC play by Euripides, a chorus of satyrs offers comic relief from the grisly story of how Polyphemus is punished for his impious behaviour in not respecting the rites of hospitality. In this play, Polyphemus claims to be a pederast, revealing to Odysseus that he takes more pleasure in boys than in women, and tries to take the satyr Silenus, whom he kept together with his sons as slaves on Mount Etna in Sicily, calling him "my Ganymede". The scene is infused with low comedy, specifically from the chorus, and Polyphemus is made to look silly: he is drunk when he explains his sexual desire, Silenus is too old to play the part of the young lover, and he himself will be subjected to penetration—with the wooden spike.

In his Latin epic, Virgil describes how Aeneas observes blind Polyphemus as he leads his flocks down to the sea. They have encountered Achaemenides, who re-tells the story of how Odysseus and his men escaped, leaving him behind. The giant is described as descending to the shore, using a "lopped pine tree" as a walking staff. Once Polyphemus reaches the sea, he washes his oozing, bloody eye socket and groans painfully. Achaemenides is taken aboard Aeneas's vessel and they cast off with Polyphemus in chase. His great roar of frustration brings the rest of the Cyclopes down to the shore as Aeneas draws away in fear.

The blinding of Polyphemus and the ruse by which Odysseus and his men escaped his cave are the only two scenes from the Odyssey shown on vases surviving from the seventh century BC (later pottery shows a much wider variety of scenes from the Odyssey). One such episode, on a vase featuring the hero carried beneath a sheep, was used on a 27 drachma Greek postage stamp in 1983.

The blinding was depicted in life-size sculpture, including a giant Polyphemus, in the Sperlonga sculptures probably made for the Emperor Tiberius. This may be an interpretation of an existing composition, and was apparently repeated in variations in later Imperial palaces by Claudius, Nero and at Hadrian's Villa.