  Motivated by symplectic geometry, we give a detailed account of differential
forms and currents on orbifolds with corners, the pull-back and push-forward
operations, and their fundamental properties. We work within the formalism
where the category of orbifolds with corners is obtained as a localization of
the category of \'etale proper groupoids with corners. Constructions and proofs
are formulated in terms of the structure maps of the groupoids, avoiding the
use of orbifold charts. The Fr\'echet space of differential forms on an
orbifold and the dual space of currents are shown to be independent of which
\'etale proper groupoid is chosen to represent the orbifold.
