This thesis deals with the study of the polynomial identities of matrices. A well-known problem of Procesi is to find the dimension of the ideal of identities of M(,n)(k) in a fixed degree. Together with Formanek and Li a partial solution to this problem is given by determining the Poincare series of the ideal of two-variable identities of M(,2)(k). The weak identities of M(,n)(k) are those polynomials which vanish on sl(,n)(k). The Poincare series of the ideal of two-variable weak identities of M(,2)(k) is determined. Also the Poincare series of the ideal of n-variable identities of upper triangular 2 x 2 matrices is found. As an application this ideal is determined explicitly. The final portion of this thesis is concerned with the central polynomial identities of M(,n)(k). Razmyslov gave a technique for constructing central polynomials from weak identities. This technique is modified and simple new proofs are given. As an application a central polynomial of degree n('2) for M(,n)(k) is constructed.
