  In this paper we study string compactifications on Deligne-Mumford stacks.
The basic idea is that all such stacks have presentations to which one can
associate gauged sigma models, where the group gauged need be neither finite
nor effectively-acting. Such presentations are not unique, and lead to
physically distinct gauged sigma models; stacks classify universality classes
of gauged sigma models, not gauged sigma models themselves. We begin by
defining and justifying a notion of ``Calabi-Yau stack,'' recall how one
defines sigma models on (presentations of) stacks, and calculate of physical
properties of such sigma models, such as closed and open string spectra. We
describe how the boundary states in the open string B model on a Calabi-Yau
stack are counted by derived categories of coherent sheaves on the stack. Along
the way, we describe numerous tests that IR physics is
presentation-independent, justifying the claim that stacks classify
universality classes. String orbifolds are one special case of these
compactifications, a subject which has proven controversial in the past;
however we resolve the objections to this description of which we are aware. In
particular, we discuss the apparent mismatch between stack moduli and physical
moduli, and how that discrepancy is resolved.
