  ``An orbifold is a space which is locally modeled on the quotient of a vector
space by a finite group.'' This sentence is so easily said or written that more
than one person has missed some of the subtleties hidden by orbifolds.
Orbifolds were first introduced by Satake under the name ``V-manifold'' and
rediscovered by Thurston who called them ``orbifolds''. Both of them used only
faithful actions to define their orbifolds (these are the so called reduced
orbifolds). This intuitive restriction is very unnatural mathematically if we
want for example to study suborbifolds. Orbispaces are to topological spaces
what orbifolds are to manifolds. They have been defined by Haefliger via
topological groupoids. He defined the homotopy and (co)homology groups of an
orbispace to be those of the classifying space of the topological groupoid.
However, his definition of morphisms is complicated and technical. Ruan and
Chen have tried to reformulate the definitions, but they don't seem treat the
non-reduced case in a satisfactory way. Our idea is to define orbispaces and
orbifolds directly via their classifying spaces. This approach allows us to
easily define morphisms of orbispaces and orbifolds with all the desired good
properties.
