In this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations. Starting from the fundamental ideas, we go further on building auxilliary techniques, as the treating of boundary conditions, and presenting accessory tools like mapping and filtering, finishing with a complete algorithm to solve a classical problem of fluid dynamics: The flow through a circular obstacle. We also present a short comparison with Finite Differences, showing the superior efficiency of spectral methods in problems with smooth solutions. Several equations like the wave equation in one spatial dimension, Burgers in a 2D domain and a simple multidomain setting for the Navier-Stokes 2D are solved numerically and the results are presented at the applications sections. We end with a brief presentation on the software PseudoPack2000 and a quick discussion on the relevant literature.