  The first goal of this survey paper is to argue that if orbifolds are
groupoids, then the collection of orbifolds and their maps has to be thought of
as a 2-category. Compare this with the classical definition of Satake and
Thurston of orbifolds as a 1-category of sets with extra structure and/or with
the "modern" definition of orbifolds as proper etale Lie groupoids up to Morita
equivalence.
  The second goal is to describe two complementary ways of thinking of
orbifolds as a 2-category:
  1. the weak 2-category of foliation Lie groupoids, bibundles and equivariant
maps between bibundles and
  2. the strict 2-category of Deligne-Mumford stacks over the category of
smooth manifolds.
