Abstract: In 1975 a new trend of commutative algebra arose with the work by Richard Stanley who used the theory of Cohen-Macaulay rings to prove affirmatively the upper bound conjecture for spheres. It then turned out that commutative algebra supplies basic methods in the algebraic study of combinatorics on convex polytopes and simplicial complexes. Stanley was the first who used in a systematic way concepts and technique from commutative algebra to study simplicial complexes by considering the Hilbert function of Stanley-Reisner rings, whose defining ideals are generated by square-free monomials. Since then, the study of square-free monomial ideals from both the algebraic and combinatorial point of view is one of the most exciting topics in commutative algebra. In this talk we present a survey on some research on combinatorial commutative algebra.