As a tool for large-scale computations in fluid dynamics, spectral methods were originally proposed in 1944 by Blinova, first implemented in 1954 by Silberman, virtually abandoned in the mid-1960s, resurrected in 1969–70 by Orszag and by Eliason, Machenhauer and Rasmussen, developed for special- ized applications in the 1970s, endowed with the first mathematical foun- dations by the seminal work of Gottlieb and Orszag in 1977, extended to a broader class of problems and thoroughly analyzed in the 1980s, and en- tered the mainstream of scientific computation in the 1990s. Two decades ago when we wrote Spectral Methods in Fluid Dynamics (1988) both the subject and the authors were barely past their adolescence. As the field and the authors are now in their middle age, the time seems ripe for a more mature discussion of the field, accounting for the main contributions of the intervening years. Motivated by the many favorable comments we have re- ceived and the continuing interest in the first book (which will be referred to as CHQZ1), yet desiring to present a more modern perspective, we embarked on a project which has resulted in this book (referred to as CHQZ2) and its companion book (Canuto, Hussaini, Quarteroni and Zang (2007), referred to as CHQZ3). These, like our first text on this subject, are books about spectral methods for partial differential equations – when to use them, how to implement them, and what can be learned from their rigorous theory.