  In supersymmetric models with gravity-mediated SUSY breaking, universality of
soft SUSY breaking sfermion masses m_0 is motivated by the need to suppress
unwanted flavor changing processes. The same motivation, however, does not
apply to soft breaking Higgs masses, which may in general have independent
masses from matter scalars at the GUT scale. We explore phenomenological
implications of both the one-parameter and two-parameter non-universal Higgs
mass models (NUHM1 and NUHM2), and examine the parameter ranges compatible with
Omega_CDM h^2, BF(b --> s,gamma) and (g-2)_mu constraints. In contrast to the
mSUGRA model, in both NUHM1 and NUHM2 models, the dark matter A-annihilation
funnel can be reached at low values of tan(beta), while the higgsino dark
matter annihilation regions can be reached for low values of m_0. We show that
there may be observable rates for indirect and direct detection of neutralino
cold dark matter in phenomenologically aceptable ranges of parameter space. We
also examine implications of the NUHM models for the Fermilab Tevatron, the
CERN LHC and a Sqrt(s)=0.5-1 TeV e+e- linear collider. Novel possibilities
include: very light s-top_R, s-charm_R squark and slepton_L masses as well as
light charginos and neutralinos and H, A and H^+/- Higgs bosons.
