This book is a basic and comprehensive introduction to the use of spectral methods for the approximation of the solution to ordinary differential equations and time-dependent boundary-value problems. The algorithms are presented and studied both from the point of view of the theoreticalanalysis of convergence and the numerical implementation. Unlike other texts devoted to the subject this is a concise introduction that is ideally suited to the novice and practitioner alike, enabling them to assimilate themethods quickly and efficiently.