We also know the stories of Thetis dipping her son in the river Styx to make him nigh-invulnerable, but there is conflicting evidence.
Those who have read the Iliad may remember the incident with
the warrior Asteropaeus hurled with both spears at once, for he was one that could use both hands alike. With the one spear he smote the shield, but it brake not through, for the gold stayed it, the gift of the god and with the other he smote the right forearm of Achilles a grazing blow, and the black blood gushed forth; but the spear-point passed above him and fixed itself in the earth, fain to glut itself with flesh. Hom. Il. 21.161
But even here, there is the implication that the weapon itself was disinclined to do serious damage.
It is Euripides, the great iconoclast of Greek Drama that makes a big deal of it:
Achilles: A fearful cry is heard among the Argives.
Clytemnestra: What is it? tell me.
Achilles: It concerns your child.
Clytemnestra: An evil omen for your words.
Achilles: They say her sacrifice is necessary.
Clytemnestra: And is there no one to say a word against them?
Achilles: Indeed I was in some danger myself from the tumult.
Clytemnestra: In danger of what, stranger?.
Achilles: Of being stoned.
Euripides was surely playing off of the myth of Achilles' invincibility. In this scene, the hero who fights whole armies single-handedly feels himself in danger when faced with armed conflict with the Argives. This moment is pretty shocking to those raised with the Thetis story.
So what gives? Was Achilles nigh-invincible or not?