  Equations arising in General Relativity are usually too complicated to be
solved analytically and one has to rely on numerical methods to solve sets of
coupled partial differential equations. Among the possible choices, this paper
focuses on a class called spectral methods where, typically, the various
functions are expanded onto sets of orthogonal polynomials or functions. A
theoretical introduction on spectral expansion is first given and a particular
emphasis is put on the fast convergence of the spectral approximation. We
present then different approaches to solve partial differential equations,
first limiting ourselves to the one-dimensional case, with one or several
domains. Generalization to more dimensions is then discussed. In particular,
the case of time evolutions is carefully studied and the stability of such
evolutions investigated. One then turns to results obtained by various groups
in the field of General Relativity by means of spectral methods. First, works
which do not involve explicit time-evolutions are discussed, going from rapidly
rotating strange stars to the computation of binary black holes initial data.
Finally, the evolutions of various systems of astrophysical interest are
presented, from supernovae core collapse to binary black hole mergers.
