Summary: The goal of this paper is twofold. First, we give an elementary introduction to the usage of spectral sequences in the combinatorial setting. Second we list a number of applications. In the first group of applications the simplicial complex is the nerve of a poset; we consider general posets and lattices, as well as partition-type posets. Our last application is of a different nature: the $S _{ n}$ mathcalS_n -quotient of the complex of directed forests is a simplicial complex whose cell structure is defined combinatorially.

spectral sequences, posets, graphs, homology groups, shellability