This text provides a hands-on introduction to spectral methods in MATLAB. It is built around 40 short and powerful MATLAB programs. Users of this book include advanced undergraduate and graduate students studying numerical methods for PDEs, numerical analysts, engineers, and computationally oriented physical scientists in all areas. The reader must have reasonable exposure to linear algebra, PDEs at the practical (not theoretical) level, and MATLAB. This book presents the key ideas along with many figures, examples, and short, elegant MATLAB programs for readers to adapt to their own needs. It covers ODE and PDE boundary value problems, eigenvalues and pseudospectra, linear and nonlinear waves, and numerical quadrature. In addition, a supplemental set of MATLAB code files is available for download.