Small
clarification to the earlier writeup.
In fact Andrew Wiles was unable to prove the Taniyama-Shimura
conjecture, in general. He only established it in the semistable case (with some help from Richard Taylor), when the rational elliptic curve has a squarefree conductor. However, this special case was enough to prove Fermat's Last Theorem.
The Taniyama-Shimura conjecture itself is a more interesting
result that FLT and has now been proved in full. The result is due to
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.
Predictably, this has not aroused the same level of media excitement as Wiles' announcement of FLT...