  In recent work, the second author introduced the concept of Coxeter quivers,
generalizing several previous notions of a quiver representation. Finite type
Coxeter quivers were classified, and their indecomposable objects were shown to
be in bijection with positive roots, generalizing a classical theorem of
Gabriel. In this paper we define fusion quivers, a natural generalization of
Coxeter quivers. We classify the finite type fusion quivers, and prove the
analogue of Gabriel's theorem. As a special case, this proves a generalised
quantum McKay correspondence for fusion categories, an analogue of
Auslander--Reiten's result for finite groups in the fusion categorical setting.
