Using the presentation of orbifolds by proper etale groupoids we first explain how a deformation quantization of an orbifold induces a formal deformation of the convolution algebra. We then construct a twisted trace density, which gives rise to a kind of universal trace on the deformed convolution algebra. The algebraic index formula for orbifolds expresses this trace in terms of characteristic classes of the orbifold. In the case of a reduced orbifold, our index formula proves a conjecture by Fedosov, Schulze and Tarkhanov. Moreover, it also entails the index theorem by Kawasaki for elliptic operators on orbifolds. (Joint work with N. Neumaier, H. Posthuma and X. Tang).