I bought this book to help me understand spectral methods enough to employ them for a nonlinear PDE problem that I have been working on. I sure am glad that I did! Here is a very complete and readable account of both the theoretical underpinnings and practical aspects of spectral methods. Spectral methods offer a fantastic alternative to classical approachs for numerical solutions of PDEs (such as finite-difference). In many ways, they are vastly superior - in terms of both accuracy and computational speed. What really makes this book so nice is that very little background is assumed (a little bit of mathematical analysis and some background in numerical analysis would help) and it is very straightforward to code up (I used Matlab) the examples sprinkled throughout the text. I also liked the fact that it is short and direct-at only 273 pages and cleanly partitioned into 12 chapters which expertly guide the reader from fundamentals up through advanced topics - it is quickly digestible and useable for practical applications! As an added bonus, an annotated bibliography is provided at the end of each chapter.