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.