For a finite group 퐺 , we define an equivariant cobordism category 풞퐺푑 . Objects of the category are (푑−1) ‐dimensional closed smooth 퐺 ‐manifolds and morphisms are smooth 푑 ‐dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (that is, geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of a certain equivariant Thom spectrum.