Abstract: The cohomology of a moment-angle complex is given by the Koszul homology of the Stanley-Reisner ring of the underlying simplicial complex. The Hochster decomposition gives an easy way to compute this Koszul homology from the combinatorics of the simplicial complex. In this talk, we will discuss Hochster's proof of the decomposition. Then, we will look at some 'colorful' generalizations of the decomposition which correspond to cohomology of certain partial quotients of moment-angle complexes.