In the past few years (1995–98), I have given several advanced graduate courses at UCLA in order to provide a comprehensive account of the proof by Wiles (and Taylor) of the identification of certain Hecke algebras with universal deformation rings of Galois representations. Assuming a good knowledge of Class field theory, I started with an overview of the theory of automorphic forms on linear algebraic groups, specifically, GL(n) over number fields. Since second year graduate students often lack knowledge of representation theory of profinite groups, necessary to carry out the task, I went on to describe basic representation theory, the theory of pseudo-representations and their deformation. To reach this point, I had already covered almost a one-year course. Then I continued to give a sketch of the rationality and the control theorems of the space of elliptic modular forms, which is the basis of the definition of the Hecke algebra. In the meantime, K. Fujiwara and F. Diamond independently gave, in 1996, a substantial simplification of the proof of Wiles, which I incorporated in my course. After having proved the theorem, assuming many things, I came back to the material I used in the proof, in particular the duality theorems (due to Poitou and Tate) of Galois cohomology groups. Thus the first chapters follow faithfully my series of courses; so, logically the reader might have to jump around between chapters.