In his important papers [16], [17], Wiles proved in most cases the so-called main conjecture for the cyclotomic Zp-extension of any totally real base field F, for all odd primes p, and for all abelian characters of Gal(Q¯ /F) (we say most cases because his work only establishes the main conjecture up to µ-invariants for those abelian characters whose order is divisible by p). It would be technically too difficult for us in these introductory lectures to attempt to explain his proof in this generality. Instead, we have chosen the much more modest path of giving a sketch of his proof in the very special case that F = Q, and for all abelian characters of Q of p-power conductor. In fact, our account of Wiles’ proof in this case has been directly inspired by a series of lectures on this theme, which we attended, given by Chris Skinner in an instructional conference held at the Centre of Mathematical Sciences, Zhejiang University, Hangzhou, China, in August 2004.