Achilles dies during the Trojan War (but after the action of the Iliad
) mortally wounded by an arrow shot by Paris
. Ovid (Metamorphoses
12) has Apollo urge Paris to shoot at Achilles and then guide his aim. Other writers allow Paris to do the shooting (or stabbing) alone, or Apollo, or Apollo disguised as Paris. Apollodorus and others say the wound was in Achilles' heel. Not all the authors subscribed to the idea that Achilles was only mortal in his heel, especially since it doesn't make a lot of sense to think that an ordinary wound in the ankle would be lethal. The bronze man Talos, however, did die when the nail in his ankle was removed and all the life-giving fluid running through his body leaked out. That Achilles' mother was a nymph made Achilles a demi-god, at best. Her attempts to make him immortal by burning or immersion in the River Styx were obviously not entirely successful.
Frazer's notes to Apollodorus go through the variants and authors.