We introduce and study the almost topological groups which are fundamentally a generalization of topological groups. Almost topological groups are defined by using almost continuous mappings in the sense of Singal and Singal. We investigate some permanence properties of almost topological groups. It is proved that translation of a regularly open (resp. regularly closed) set in an almost topological group is regularly open (resp. regularly closed). And this fact gives us a lot of important and useful results of almost topological groups.