The most famous Cyclop, who allegedly lived on the island Youra in the North Sporades. His father was Poseidon and mother the nymph Thoosa.
According to Homer, Polyphemus sealed his cave with a huge stone when he discovered that Odysseus and his men had hid inside it. After eating a few of them he had a conversation with Odysseus, who told him his name was "Noone".
After getting the giant drunk, Odysseus talked his men into preparing a sharp pole by putting it in the fireplace. When it was burning hot, the men shoved into the Cyclops only eye. Blinded and suffering the pain, Polyphemus removed the entrance stone, and tried to catch the escaping men. Odysseus had foreseen this, though, and all the men had tied themselves under the sheep, so the Cyclop only felt the animals' backs as they left the cave.
Polyphemus shouted across to the other Cyclops that Noone had tried to kill him but the fellow Cyclops took him for mad, and did not pay any notice. In a last desperate attempt to kill the men, Polyphemus threw huge rocks into the sea, almost hitting Odysseus ship.
In another myth, Polyphemus was desperately in love with the mocking Galatea.