  We construct an algebra of pseudodifferential operators on each groupoid in a
class that generalizes differentiable groupoids to allow manifolds with
corners. We show that this construction encompasses many examples. The
subalgebra of regularizing operators is identified with the smooth algebra of
the groupoid, in the sense of non-commutative geometry. Symbol calculus for our
algebra lies in the Poisson algebra of functions on the dual of the Lie
algebroid of the groupoid. As applications, we give a new proof of the
Poincar\'e-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization
of the Lie-Poisson structure on the dual $A^*$ of a Lie algebroid.
