Combinatorial commutative algebra is a new branch in the modern research of mathematics. It has recently developed by combining different techniques used in commutative algebra and in combinatorics. Monomial ideals are at the intersection of commutative algebra and combinatorics. The radical monomial ideals which are ideals generated by squarefree monomials, have a beautiful combinatorial interpretation in terms of simplicial complexes. With any simplicial complex one may associate a squarefree monomial ideal and viceversa. By this process, the combinatorial properties of the simplicial complexes can be described by using algebraic methods on the Stanley-Reisner ideals, and many invariants of the squarefree monomial ideals can be studied by combinatorial methods applied to the associated simplicial complexes. In this book we are interested in describing some homological invariants of certain classes of monomial ideals. A special attention is payed to squarefree and non-squarefree lexsegment ideals, because of the major role played by them in the study of the Hilbert function.