Consider all of your basic constructions/tools/theorems for (possibly smooth) manifolds: fundamental group, Euler characteristic, triangulations, orientation, bundle structure, cobordisms, etc. Viewing orbifolds as the natural generalization of manifolds by quotients of group actions, it *seems* that we can translate all of our notions for manifolds onto orbifolds with slight adjustments in the definitions. I am curious as to the extent of this (which is important, because orbifolds are arising everywhere for me):  
**What basic constructions that exist on *manifolds*, cannot (or currently do not) have analogs for *orbifolds*?**  
Why? ($\leftarrow$ in case there is more to say than "orbifolds have singularities")