  Manifolds without boundary, and manifolds with boundary, are universally
known in Differential Geometry, but manifolds with corners (locally modelled on
[0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic
definitions in the subject are not agreed upon, there are several inequivalent
definitions in use of manifolds with corners, of boundary, and of smooth map,
depending on the applications in mind.
  We present a theory of manifolds with corners which includes a new notion of
smooth map f : X --> Y. Compared to other definitions, our theory has the
advantage of giving a category Man^c of manifolds with corners which is
particularly well behaved as a category: it has products and direct products,
boundaries behave in a functorial way, and there are simple conditions for the
existence of fibre products X x_Z Y in Man^c.
  Our theory is tailored to future applications in Symplectic Geometry, and is
part of a project to describe the geometric structure on moduli spaces of
J-holomorphic curves in a new way. But we have written it as a separate paper
as we believe it is of independent interest.
