The representation of spatial derivatives is at the heart of spectral methods for partial differential equations, so the main kinds (Fourier, Chebyshev, Legendre, Laguerre, Hermite) are analyzed at the outset, together with efficient means to compute them. Galerkin methods involving all three classes of basis functions are discussed for both stationary (Helmholtz equation) and non-stationary (advection equation) problems. Tau methods are also applicable to both types of problems and are presented next. Due to their straightforward implementation of boundary conditions, they offer an exciting alternative to Galerkin approaches. Separate Sections are devoted to collocation methods and efficient means to solve non-linear equations in the spectral framework. Strong stability preserving time integration methods are discussed, and spectral methods for ODE and PDE posed on semi-infinite and infinite definition domains are introduced. Examples and Problems include the Galerkin method for the advection, diffusion and Poisson equations (Poiseuille flow), the tau method for the Poisson equation, the diffusion equation in collocation approaches, and the Burgers equation.