Introduction Wavelets are a relatively recent .evelopment in applied mathematics. Their name itself was coined approximately a decade ago(Morlet, Arens, Fourgeau and Giard(1982), Morlet(1983), Grossmann and Morlet( 1984); in the last ten years interest in them has grown at an explosive rate. There are several rea- sons for their present success. On the one hand the concept of wavelets can bi viewed as a synthesis of ideas which originated during the last twenty or thirty years in engineering(subband coding), physics(coherent states, renormalization group), and pure mathematics(study of Calderon-Zygmund operators).As a consequence of these interdisciplinary origins, wavelets appeal to scientists and engineers of many different backgrounds. On the other hand, wavelets are a fairly simple mathematical tool with a great, variety of possible applications. Already they have led to exciting applications in signal analysis(sound, images)(some early references are Kronland-Martinet, Morlet and Grossmann(1987), Mallat (1989b),(1989c); more recent references are given later)and numerical analy- sis(fast algorithms for integral transforms in Beylkin, Coifman, and Rokhlin (1991)); many other applications are being studied. This wide applicability alse contributes to the interest they generate V This book contains ten lectures I delivered as the principal speaker at the CBMS conference on wavelets organized in June 1990 by the Mathematics De- partment at the University of Lowell, Massachusetts. according to the usual format or the CBMS conferences, other speakers(G. Battle, G. Beylkin, C. Chui A. Cohen, R. Coifman, K. Grochenig, J. Liandrat, S. Mallat, B. Torresani and A. Willsky) provided lectures on their work related to wavelets. Moreover three workshops were organized, on applications to physics and inverse problems (chaired by B. DeFacio), group theory and harmonic analysis(H. Feichtinger and signal analysis(M. Vetterli). The audience consisted of researchers active in the field of wavelets as well as of mathematicians and other scientists and ngineers who knew little about wavelets and hoped to learn more. This second group constituted the largest part of the audience I saw it as my task to provide a tutorial on wavelets to this part of the audience, which would then be a solid grounding for more recent work exposed by the other lecturers and myself. Con sequently, about two thirds of my lectures consisted of "basic wavelet theory