In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on
Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was
the prediction that a certain physicist's Euler number of the orbifold must be
equal to the Euler number of any of its crepant resolutions. This was soon
related to the so called McKay correspondence in mathematics (cf. [McK]). Later
developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of
Calabi-Yau orbifolds (cf. [Ro]), and most recently the Gromov-Witten invariants
of symplectic orbifolds (cf. [CR1-2]). One common feature of these studies is
that certain contributions from singularities, which are called ``twisted
sectors'' in physics, have to be properly incorporated. This is called the
``stringy aspect'' of an orbifold (cf. [R]).
This paper makes an effort to understand the stringy aspect of orbifolds in
the realm of ``traditional mathematics''.Comment: latex, 59 pages, minor mistakes corrected, more references adde
