Besides the usual techniques, such as finite difference methods and finite element methods, spectral methods are another important tool for numerically solving partial differential equations. If the solutions are sufficiently smooth these methods yield spectral accuracy and hence one can achieve a faster convergence rate than with any local polynomial based interpolation method. In this talk the general framework of spectral methods, i.e., collocation methods based on trigonometric interpolation, is introduced. Example nonlinear partial differential equations are solved numerically.