  This proceeding is intended to be a first introduction to spectral methods.
It is written around some simple problems that are solved explicitly and in
details and that aim at demonstrating the power of those methods. The
mathematical foundation of the spectral approximation is first introduced,
based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev
polynomials are then presented. The next section is devoted to one dimensional
equation solvers using only one domain. Three different methods are described.
Techniques using several domains are shown in the last section of this paper
and their various merits discussed.
