  For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie
algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of
$G$ (as defined by A. Connes) is shown to be a strict quantization (in the
sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's
quantization prescription on flat space. Many other known strict quantizations
are a special case of this procedure; on a Riemannian manifold, one recovers
Connes' tangent groupoid as well as a recent generalization of Weyl's
prescription. When $G$ is the gauge groupoid of a principal bundle one is led
to the Weyl quantization of a particle moving in an external Yang-Mills field.
In case that $G$ is a Lie group (with Lie algebra $g$) one recovers Rieffel's
quantization of the Lie-Poisson structure on $g^*$. A transformation group
C*-algebra defined by a smooth action of a Lie group on a manifold $Q$ turns
out to be the quantization of the semidirect product Poisson manifold $g^*x Q$
defined by this action.
