  We consider four notions of maps between smooth C^r orbifolds O, P with O
compact (without boundary). We show that one of these notions is natural and
necessary in order to uniquely define the notion of orbibundle pullback. For
the notion of complete orbifold map, we show that the corresponding set of C^r
maps between O and P with the C^r topology carries the structure of a smooth
C^\infty Banach (r finite)/Frechet (r=infty) manifold. For the notion of
complete reduced orbifold map, the corresponding set of C^r maps between O and
P with the C^r topology carries the structure of a smooth C^\infty Banach (r
finite)/Frechet (r=infty) orbifold. The remaining two notions carry a
stratified structure: The C^r orbifold maps between O and P is locally a
stratified space with strata modeled on smooth C^\infty Banach (r
finite)/Frechet (r=infty) manifolds while the set of C^r reduced orbifold maps
between O and P locally has the structure of a stratified space with strata
modeled on smooth C^\infty Banach (r finite)/Frechet (r=infty) orbifolds.
Furthermore, we give the explicit relationship between these notions of
orbifold map. Applying our results to the special case of orbifold
diffeomorphism groups, we show they inherit the structure of C^\infty Banach (r
finite)/Frechet (r=infty) manifolds. In fact, for r finite they are topological
groups, and for r=infty they are convenient Frechet Lie groups.
