Recently, Ma et al. have introduced the notions of a C∗-algebra valued metric space and C∗-algebra value contractive mappings. In this dissertation we generalize this new notion of C∗-valued contractive mappings by weakening their introduced contractive conditions in the setting of C∗-algebra valued metric spaces. Using the new notion of C∗-valued contractive type mappings, we establish some fixed point theorems for such mappings. Our result generalizes the result by Ma et al. and those contained therein except for the uniqueness. We provide an existence result for an integral equation as an application of C∗-valued contractive type mappings on complete C∗-valued metric spaces. Moreover, in the setting of C∗- algebra valued b-metric spaces, we generalize the Banach contraction principle and establish a fixed point result for a C∗-algebra valued complete b-metric spaces. Finally, for the multivalued mappings, the thesis also introduces the concept of a C∗-algebra valued metric defined on sets and then extends the result of Nadler in this setting.