From the introduction: "This article is intended to be a survey of results on modular forms and modular curves. In our attempt, and failure, to keep the work a reasonable length, we choose to ignore many important aspects of the theory and instead to emphasize those which play a role in the work of Ribet and Wiles. None of the results we present here are ours, and we have no doubt often failed to properly attribute them. We apologize in advance for these and other shortcomings, which are due largely to our ignorance. We can hardly claim to be experts on many of the topics we include; indeed we learned a great deal in preparing this article.

"We have aimed the article at advanced or recent graduate students specializing in the field, though we hope that others will find it a useful reference. Parts of the paper vary in the amount of background assumed. Beginning with $§8$, we usually take for granted graduate courses in number theory and algebraic geometry.

"The article is divided into three parts. Part I is a rapid introduction to modular forms, focusing on the theory of Hecke operators and newforms. More detailed treatments of most of the topics we cover can be found in a number of valuable texts.

"In Part II, we turn our attention to modular curves. We begin with their description as Riemann surfaces and moduli-theoretic interpretation. Then we go on to explain some of the algebraic geometric methods used to study their arithmetic and that of their Jacobians. Much of the material can be found in the work of P. Deligne and M. Rapoport \ref[in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 143--316, Lecture Notes in Math., 349, Springer, Berlin, 1973; MR 49 #2762], but much is scattered in the literature.

"Part III returns to the subject of modular forms from a more sophisticated point of view. We first give a brief introduction to modular forms in the context of automorphic representations, mainly following H. Jacquet and R. P. Langlands \ref[ Automorphic forms on ${\rm GL(2)$}, Lecture Notes in Math., 114, Springer, Berlin, 1970; MR 53 #5481]. Then we approach the subject from the perspective of the geometry of modular curves, often following G. Shimura\ \ref[ Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo, 1971; MR 47 #3318] and Deligne and Rapoport \ref[op. cit.]."