  We review the properties of transversality of distributions with respect to
submersions. This allows us to construct a convolution product for a large
class of distributions on Lie groupoids. We get a unital involutive algebra
$\cE\_{r,s}'(G,\Omega^{1/2})$ enlarging the convolution algebra
$C^\infty\_c(G,\Omega^{1/2})$ associated with any Lie groupoid $G$. We prove
that $G$-operators are convolution operators by transversal distributions. We
also investigate the microlocal aspects of the convolution product. We give
conditions on wave front sets sufficient to compute the convolution product and
we show that the wave front set of the convolution product of two distributions
is essentially the product of their wave front sets in the symplectic groupoid
$T^*G$ of Coste-Dazord-Weinstein. This also leads to a subalgebra
$\cE\_{a}'(G,\Omega^{1/2})$ of $\cE\_{r,s}'(G,\Omega^{1/2})$ which contains for
instance the algebra of pseudodifferential $G$-operators and a class of Fourier
integral $G$-operators which will be the central theme of a forthcoming paper.
