Abstract. We determine the extent to which the collection of -Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the -Euler-Satake characteristics corresponding to free or free abelian and are not classified by those corresponding to any finite set of finitely generated discrete groups. These results demonstrate that the -Euler-Satake characteristics corresponding to free abelian constitute new invariants of orbifolds. Similarly, we show that such a classification is neither possible for non-orientable 2-orbifolds nor for non-effective 2-orbifolds using any collection of groups .2010 Mathematics Subject Classification. Primary 57R20, 57S17; Secondary 22A22, 57P99.