The systematic study of prime ideals in noncommutative rings satisfying polynomial identities lends itself to diverse methods of approach. The geometric method, which we explore at several points of this thesis, was initiated by C. Procesi in the 1960's. The main results of our chapter on Prime Ideals in Affine Algebras are most naturally developed from this point of view. They lead to certain algebraic questions which are dealt with in this thesis, concerning the structure of prime ideals in affine algebras satisfying polynomial identities. The interplay between the prime ideals in such as algebra, and the center of the algebra is a theme that is exploited frequently, through the use of central polynomials. We also answer a question posed by Procesi concerning the growth of affine, prime algebras satisfying polynomial identities.