Even though there is a wide range of numerical methods to treat partial differential equations (PDEs), our interests lie in those which render highly accurate solutions, ideally close to machine precision. In this context, pseudo-spectral methods are probably the best choice, as they have the remarkable capability of providing exponential convergence rate when the underlying problem admits a regular solution. Therefore, the aim of this talk is to give an introductory discussion to such spectral-methods applied to some problems in classical field theories. I shall first concentrate on examples concerning PDEs restricted to spatial coordinates (typically elliptic equations), a field where spectral methods find their most common application. Then I will discuss our efforts to further extend those methods to the realm of hyperbolic equations, where a spectral approximation is also considered in the time direction.
