From Wikipedia’s entry on Achilles:
The death of Achilles, as predicted by Hector with his dying breath, was brought about by Paris with an arrow (to the heel according to Statius). In some versions, the god Apollo guided Paris’s arrow. Some retellings also state that Achilles was scaling the gates of Troy and was hit with a poisoned arrow.
All of these versions deny Paris any sort of valor, owing to the common conception that Paris was a coward and not the man his brother Hector was and Achilles remained undefeated on the battlefield. His bones were mingled with those of Patroclus, and funeral games were held. He was represented in the Aethiopis as living after his death in the island of Leuke at the mouth of the river Danube.
Another version of Achilles’s death is that he fell deeply in love with one of the Trojan princesses, Polyxena. Achilles asks Priam for Polyxena’s hand in marriage. Priam is willing because it would mean the end of the war and an alliance with the world’s greatest warrior. But while Priam is overseeing the private marriage of Polyxena and Achilles, Paris, who would have to give up Helen if Achilles married his sister, hides in the bushes and shoots Achilles with a divine arrow, killing him.
Achilles was cremated and his ashes buried in the same urn as those of Patroclus.
Paris was later killed by Philoctetes using the enormous bow of Heracles.