  We develop a new method to compute the homology groups of finite topological
spaces (or equivalently of finite partially ordered sets) by means of spectral
sequences giving a complete and simple description of the corresponding
differentials. Our method proves to be powerful and involves far fewer
computations than the standard one. We derive many applications of our
technique which include a generalization of Hurewicz theorem for regular
CW-complexes, results in homological Morse theory and formulas to compute the
M\"obius function of posets.
