Abstract: Orbifolds are useful generalizations of manifolds. In this talk I will present a framework to study cobordism of orbifolds. As application, I will show calculations of several cobordism groups of orbifolds in terms of usual smooth bordism theory, these decompositions involve information around the singular sets and provide a way to define invariants for orbifolds. These groups form the coefficients of a new generalized homology theory. Orbi-bordism.