In this paper, we initiate the study of generalized topological groups. A generalized topological group has the algebraic structure of groups and the topological structure of a generalized topological space defined by \'A. Cs\'asz\'ar  and they are joined together by the requirement that multiplication and inversion are $\Cal G$-continuous. Every topological group is a $\Cal G$topological group whereas converse is not true in general. Quotients of generalized topological groups are defined and studied.