  This is the first of a series of papers which are devoted to a comprehensive
theory of maps between orbifolds. In this paper, we define the maps in the more
general context of orbispaces, and establish several basic results concerning
the topological structure of the space of such maps. In particular, we show
that the space of such maps of C^r-class between smooth orbifolds has a natural
Banach orbifold structure if the domain of the map is compact, generalizing the
corresponding result in the manifold case. Motivations and applications of the
theory come from string theory and the theory of pseudoholomorphic curves in
symplectic orbifolds.
