  Giving up the assumption of the gaugino mass unification at the GUT scale,
the latest LEP and Tevatron data still allow the lightest supersymmetric Higgs
to have a large branching fraction into invisible neutralinos. Such a Higgs may
be difficult to discover at the LHC and is practically unreachable at the
Tevatron. We argue that, for some of these models to be compatible with the
relic density, light sleptons with masses not far above the current limits are
needed. There are however models that allow for larger sleptons masses without
being in conflict with the relic density constraint. This is possible because
these neutralinos can annihilate efficiently through a Z-pole. We also find
that many of these models can nicely account, at the 2\sigma level, for the
discrepancy in the latest g-2 measurement. However, requiring consistency with
the g-2 at the 1\sigma level, excludes models that lead to the largest Higgs
branching fraction into LSP's. In all cases one expects that even though the
Higgs might escape detection, one would have a rich SUSY phenomenology even at
the Tevatron, through the production of charginos and neutralinos.

High Energy Physics - Phenomenology