This text presents the modern unified theory of spectral methods and their implementation in the numerical analysis of partial differential equations occurring in fluid dynamical problems of transition, turbulence and aerodynamics. It provides the engineer with the tools and guidance necessary to apply the methods successfully, and it furnishes the mathematician with a theoretical background to the subject. All of the essential components of spectral algorithms currently employed for large-scale computations in fluid mechanics are described in detail. Some specific applications are linear stability, boundary layer calculations, direct simulations of transition and turbulence, and compressible Euler equations. The authors also present complete algorithms for Poisson's equation, linear hyperbolic systems, the advection diffusion equation, isotropic turbulence, and boundary layer transition. Some recent developments stressed in the book are iterative techniques (including the spectral multigrid method), spectral shock-fitting algorithms, and spectral multidomain methods. The book addresses graduate students and researchers in fluid dynamics and applied mathematics as well as engineers working on problems of practical importance. For this edition the authors have added a new section on the spectral domain decomposition method, and a supplementary bibliography on recent developments.