Abstract: 
The aim of this paper is to introduce a spectral sequence that converges to the cobordism groups of orbifolds with given isotropy representations. In good cases the $E^1$-term of this spectral sequence is given by a certain cobordism group of orbibundles over purely ineffective orbifolds which can be identified with the bordism group of the classifying space of the Weyl group of a finite subgroup of $O(n)$. We use this spectral sequence to calculate some cobordism groups of orbifolds for low dimensions, and in particular, we show that every three dimensional effective oriented orbifold, or even only locally oriented orbifold, bounds. And although every two dimensional effective oriented or\-bi\-fold bounds, $\mathbb{RP}^2$ is the generator of the second cobordism group of locally oriented orbifolds.

1. Department of Mathematics
   Stanford University
   Stanford, CA 94305, U.S.A.