The purpose of this paper is to introduce and study the notions of [Formula: see text]-rack and [Formula: see text]-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order [Formula: see text]-quandles. Moreover, we define modules over [Formula: see text]-racks, discuss extensions and define a cohomology theory for [Formula: see text]-quandles and give examples.