Abstract: The study of enumerative properties of simplicial complexes dates back to the Euler-Poincare formula. The groundbreaking work of Hochster, Reisner and Stanley in the 1970's established the combinatorial properties of Cohen-Macaulay complexes. First we will describe what these complexes are and indicate why they are important. Then we will examine what happens if we ask for stronger restrictions on the geometry of the spaces. Finally, we will see what can be said if we relax some of the conditions, as is required in order to study triangulations of manifolds.