In this paper we study the class of s-topological groups and a wider class of S-topological groups which are defined by using semi-open sets and semi-continuity introduced by N. Levine. It is shown that these groups form a generalization of topological groups, and that they are different from several distinct notions of semitopological groups which appear in the literature. Counterexamples are given to strengthen these concepts. Some basic results and applications of s- and S-topological groups are presented. Similarities with and differences from topological groups are investigated.