  We give a notion of equivalence for Fell bundles over groups, not necessarily
saturated nor separable, and show that equivalent Fell bundles have
Morita-Rieffel equivalent cross-sectional $C^*$-algebras. Our notion is
originated in the context of partial actions and their enveloping actions. The
equivalence between two Fell bundles is implemented by a bundle of Hilbert
bimodules with some extra structure. Suitable cross-sectional spaces of such a
bundle turn out to be imprimitivity bimodules for the cross-sectional
$C^*$-algebras of the involved Fell bundles. We show that amenability is
preserved under this equivalence and, by means of a convenient notion of
internal tensor product between Fell bundles, we show that equivalence of Fell
bundles is an equivalence relation.
