Spectral methods represent a family of methods for the numerical approximation of partial differential equations. Their common denominator is to rely on high-order polynomial expansions, notably trigonometric polynomials for periodic problems, and orthogonal Jacobi polynomials for nonperiodic boundary-value problems. They have the potential of providing a high rate of convergence when applied to problems with regular data. They can be regarded as members of the broad family of (generalized) Galerkin methods with numerical evaluation of integrals based on Gaussian nodes. In the first part, we introduce the methods on a computational domain of simple shape and analyze their approximation properties as well as their algorithmic features. Next, we address the issue of how these methods can be extended to more complex geometrical domains by retaining their distinctive approximation properties.