In this paper, we present some contractive mappings and prove new generalized fixed point theorems on $S$-metric spaces. Also we define the notion of a cluster point and investigate fixed points of self-mappings using cluster points on $S$-metric spaces. We obtain new generalizations of the classical Nemytskii-Edelstein and Ćirić's fixed point theorems for continuous self-mappings of compact $S$-metric spaces.