  We introduce a framework of translation quiver varieties which includes
Nakajima quiver varieties as well as their graded and cyclic versions. An
important feature of translation quiver varieties is that the sets of their
fixed points under toric actions can be again realized as translation quiver
varieties. This allows one to simplify quiver varieties in several steps. We
prove that translation quiver varieties are smooth, pure and have Tate motivic
classes. We also describe an algorithm to compute those motivic classes.
