  The paper establishes, for a wide class of locally compact groupoids
$\Gamma$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a
way parallel to that established for locally compact groups by Guentner, Higson
and Trout. The second section shows that $\Gamma$-actions on a
$C_{0}(X)$-algebra $B$, where $X$ is the unit space of $\Gamma$, can be
usefully formulated in terms of an action on the associated bundle
$B^{\sharp}$. The third section shows that the functor $B\to C^{*}(\Gamma,B)$
is continuous and exact, and uses the disintegration theory of J. Renault. The
last section establishes the existence of the descent functor under a very mild
condition on $\Gamma$, the main technical difficulty involved being that of
finding a $\Gamma$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the
group case.
