  We consider the production of supersymmetry neutralinos and charginos in
$p\bar p$ collisions at the Tevatron, and their subsequent decay via
hadronically quiet dileptons and trileptons. We perform our computations in the
context of a variety of supergravity models, including generic four-parameter
supergravity models, the minimal $SU(5)$ supergravity model, and $SU(5)\times
U(1)$ supergravity with string inspired two- and one-parameter moduli and
dilaton scenarios. Our results are contrasted with estimated experimental
sensitivities for dileptons and trileptons for integrated luminosities of
$100\ipb$ and $1\ifb$, which should be available in the short and long terms.
We find that the dilepton mode is a needed complement to the trilepton signal
when the latter is suppressed by small neutralino leptonic branching ratios.
The estimated reaches in chargino masses can be as large as $100\,(150)\GeV$
for $100\ipb\,(1\ifb)$. We also discuss the task left for LEPII once the
Tevatron has completed its short-term search for dilepton and trilepton
production.
