Consider all of your basic constructions/tools/theorems for manifolds: fundamental group, Euler characteristic, triangulations, orientation, smoothness, bundle structure, cobordisms, etc.. Viewing orbifolds as the natural generalization of manifolds by quotients of group actions (or just locally $\mathbb{R}^n/G_i$), 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 that statement's validity (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")

As pointed out nicely below, there is the question of whether such constructions will be *useful* to (all) orbifolds. So this thread also seeks known constructions which are either intractable or trivial.