As a generalization of positive implicative $BCK$-algebra, the notion of generalized $BCK$-algebras is introduced. A method to make a $gBCK$-algebra from a quasi-ordered set is provided. The notion of generalized $BCK$-ideals of generalized $BCK$-algebras is introduced, and then the connections between such ideals and congruences are considered. Characterizations of generalized $BCK$-ideals are given. A generalized $BCK$-ideal generated by a set is established.