  In this article we introduce the notion of fundamental solution in the
Colombeau context as an element of the dual $\LL(\Gc(\R^n),\wt{\C})$. After
having proved the existence of a fundamental solution for a large class of
partial differential operators with constant Colombeau coefficients, we
investigate the relationships between fundamental solutions in
$\LL(\Gc(\R^n),\wt{\C})$, Colombeau solvability and $\G$- and
$\Ginf$-hypoellipticity respectively.
