The purpose of this paper is to establish some new results on the existence of fixed points for some classes of mappings of contractive type. The paper consists of three sections. The first section extends the results of Banach, Rakotch, Boyd-Wong, Mei-Keeler, Edelstein and Sehgal to singlevalued mappings of contractive type. The second section extends an earlier result of Smithson to multivalued mappings of contractive type. In the last section we extend some results of Wong, Assad-Kirk and the others to multivalued generalized contractions.