  The topic of these notes could be easily expanded into a full one-semester
course. Nevertheless, we shall try to give some flavour along with theoretical
bases of spectral and pseudo-spectral methods. The main focus is made on
Fourier-type discretizations, even if some indications on how to handle
non-periodic problems via Tchebyshev and Legendre approaches are made as well.
The applications presented here are diffusion-type problems in accordance with
the topics of the PhD school.
