According to Hesiod (Theogony, 139-146) Gaia (“Earth”) laid with with Uranus (“Sky”) and produced the “overbearing in spirit” Cyclopes. Their names were Brontes ("thunderer"), Steropes ("lightning") and Arges ("bright"). They had a single eye in the middle of their forehead, they were strong and stubborn but they were very skilled blacksmiths and made thunder and thunderbolt for Zeus. According to Hesiod (Theogony, 492-505), once Zeus had grown up, Gaia (“Earth”) forced Cronos to vomit Zeus’ siblings which Cronos previously swallowed. This happened in reverse order starting from the stone, which was set down at Pytho under the glens of Mount Parnassus to be a sign to mortal men, then his two brothers and three sisters. Then Zeus freed the brothers of his father (these would be the Cyclopes) which Cronos imprisoned in Tartarus. These, in exchange for his kindness, gave Zeus thunder, glowing thunderbolt and lightning.
According to Pseudo-Apollodorus (Biblioteca, 1.1.2) Earth coupled with Sky and gave him the Cyclopes: Arges, Steropes and Brontes. Each had one eye in the forehead. However, Sky bound them in Tartarus, a “gloomy place in Hades as far distant from earth as earth is distant from the sky”. According to Pseudo-Apollodorus (Bibliotheca, 1.2.1) when Zeus had grown up he sought the help of Metis, daughter of Oceanus (“Ocean”), who gave Cronus an emetic to force him to disgorge the children he had previously swallowed. Zeus then waged a ten-year war against Cronus and the Titans. He first freed the Hecatoncheires after slewing Campe (the female dragon that was given the task by Cronus to guard them). Then he freed the Cyclopes who gave thunder, lightning and thunderbolt to Zeus, a helmet (Helm of darkness) to Hades (“Pluto”) and a trident to Poseidon. Armed with these weapons the gods overcame the Titans, imprisoned them in Tartarus and appointed the Hecatoncheires (“Hundred-handers”) as their guards. Zeus was given the dominion of the sky, Poseidon was given the dominion of the seas and Hades (“Pluto”) was given the dominion of Hades (underworld).
The accounts of the Cyclopes by Hesiod and Pseudo-Apollodorus are very similar but the ones of Homer are quite distinct.
According to Homer (Odyssey, Book IX 105-565) Odysseus and his comrades disembarked on the island of the Cyclopes. Odysseus picked twelve of his best men and entered a cave. When inside, they gazed in wonder at all the things they found there, finding cheese and lambs. Despite the recommendations of his comrades to grab the cheese and leave right away, Odysseus did not listen because he was curious to see the man who lived in the cave. Odysseus and his man started a fire and the man came back with his flocks, he left some of the anymals outside and moved a giant rock in order to close the entrance of his cave. After starting the fire the man noticed Odysseus and his men. He questioned them with a deep voice and Odysseus explained they were Acheans coming back from the war of Troy, asking for hospitality in the name of Zeus. The tall man replied with disrespect saying that the Cyclopes did not care about Zeus or any other gods and that they were better than all of them. Odysseus explained that his ships where brought to his land because of Poseidon but the Cyclops did not reply. Instead, he grabbed two men, killed them and made his supper out of them. He drank milk and then he laid among his sheep. Then Odysseus, suddenly possessed by a fit of rage planned revenge with his sword but a second thought put him in check and made him realize that, alone, he and his men would have never been able to move the giant rock to escape from the cave. Therefore, wailing for their situation, they waited until dawn when the Cyclops woke up, started a fire, milked his flocks, grabbed two other men and used them for his next meal. He then left the cave with his animals and blocked the entrance of the cave with the giant mass, leaving Odysseus behind to devise a plan to revenge his perished comrades with the blessing of Athena. Odysseus noticed a club made of “green olive-wood” and came up with an idea. He cut one end instructed his men to make it pointy, smooth, sharpen it and burn it in the blazes of the fire. Then he hid the wood under piles of dung and told his comrades to use it to grind it in the eye of the Cyclops after he would fall asleep. After his return the Cyclops once again grabbed two men and made his supper with them. Odysseus at that point offered wine to the Cyclops who drank with pleasure and asked for more. Odysseus gave him wine a second time and a third time until the wits of the Cyclops had vanished. He then said “Noman is my name, Noman do they call me—my mother and my father, and all my comrades as well” (367). The Cyclops replied “Noman will I eat last among his comrades, and the others before him; this shall be thy gift” (369). As he spoke, he fell on his back and after a while, in his sleep, he vomited the wine and the human flesh he ate. Meanwhile, Odysseus put the stake in the ashes until it got hot and cheered his comrades. The stake was about to catch fire so Odysseus’ men took it and thrusted into the eye of the sleeping Cyclops (383) (interestingly, Homer at line 389 writes “eyelids” and “brows”). The Cyclops screamed and got up, removed the stake from his bleeding eye and flung it away from himself. The Cyclopes that lived near him came to ask what was the matter, referring to him as Polyphemus. The Cyclops at that point replied “My friends, it is Noman that is slaying me” (408) so his neighbors replied “If, then, no man does violence to thee in thy loneliness, sickness which comes from great Zeus thou mayest in no wise escape. Nay, do thou pray to our father, the lord Poseidon” (410-412). The Cyclopes left, Odysseus laughed, Polyphemus groaned in anguish and moved the stone away from the entrance to the cave, sat down with his hands stretched out in hope of catching any of the Acheans who may try to run away. Odysseus than assigned a man to a group of three sheep and he grabbed a big ram by its belly waiting for dawn. In the early morning the flocks found their way outside of the cave and Polyphemus patted the back of his animals. Outside of the cave Odysseus and his comrades ran with the sheep to their boats and they immediately set sail. At that point Odysseus screamed to Polyphemus accusing of his cruel deeds. In response, the Cyclops “broke off the peak of a high mountain” and threw it toward the ship of the Acheans who started to row away but immediately reproached Odysseus for his reckless courage. Then, a little more far in the distance Odysseus screamed: “Cyclops, if any one of mortal men shall ask thee about the shameful blinding of thine eye, say that Odysseus, the sacker of cities, blinded it, even the son of Laertes, whose home is in Ithaca” (502-505). Polyphemus then admitted that a seer name Telemus, son of Eurymus, a long time ago prophesized that the Cyclops would lose his eyesight by the hands of Odysseus and prayed to his father Poseidon that Odysseus be cursed to never reach his home in Ithaca, that he lose his friends and find enemies in his own house. After that the Cyclopes threw an even bigger rock to the ship of Odysseus. Later the flocks were divided and Odysseus sacrificed his ram to Zeus, son of Cronos, feasted on meat and wine, eventually resting with his crew after dark. At dawn the Acheans sailed on, happy to have escaped death but sad for the loss of some of their comrades.
According to Diodorus Siculus (Bibliotheca historica, 4.71.1-3) Asclepius was the son of Apollo and Coronis. He excelled in science, healing and made several discoveries. It is believed that he brought back many people from the dead. Therefore, Hades accused him in front of Zeus because the souls going to Hades were less and less. Zeus, indignated, slew Asclepius with his thunderbolt so Apollo, in turn indignated, murdered the Cyclopes who forged the thunderbolt for Zeus. At that point Zeus ordered Apollo to serve a human as a punishment for his crime.
According to Hyginus (Fabulae, 49) Asclepius, the son of Apollo, resurrected Glaucus, son of Minos (some say it was Hippolytus who was resurrected). Jupiter struck Asclepius down with a thunderbolt as a consequence. Apollo, since he could not hurt Jupiter, killed the Cyclopes who made the thunderbolt for Zeus. Therefore Zeus punished Apollo and forced him to serve King Admetus of Thessaly as a slave.
According to Pseudo-Apollodorus (Bibliotheca, 3.10.3-4) Asclepius, the son of Apollo, learned the art of medicine from Chiron the Centaur, became a god of medicine and brought people back from the dead. Because of this Zeus killed Asclepius with a thunderbolt. This angered Apollo who in turn killed the Cyclopes that made the thunderbolt for Zeus. Zeus would have hurled Apollo into Tartarus but, thanks to the intervention of Leto, Zeus ordered Apollo to serve Admetus as a herdsman for a year.