The aim of this paper is to find a generalization of topological groups. The concept arises out of the investigation to obtain a group structure on the set $[X,Y]$, of homotopy classes of maps from a space $X$ to a given space $Y$ for all $X$ which is natural with respect to $X$. We also study the generalized topological groups. Finally, associated with each generalized topological group we construct a contravariant functor from the homotopy category of pointed topological spaces and base point preserving continuous maps to the category of groups and homomorphisms.