  We discuss the possibilities for the direct detection of neutralino dark
matter via elastic scattering in variants of the minimal supersymmetric
extension of the Standard Model (MSSM) with non-universal
supersymmetry-breaking contributions to the Higgs masses, which may be either
equal (NUHM1) or independent (NUHM2). We compare the ranges found in the NUHM1
and NUHM2 with that found in the MSSM with universal supersymmetry-breaking
contributions to all scalar masses, the CMSSM. We find that both the NUHM1 and
NUHM2 offer the possibility of larger spin-independent dark matter scattering
cross sections than in the CMSSM for larger neutralino masses, since they allow
the density of heavier neutralinos with large Higgsino components to fall
within the allowed range by astrophysics. The NUHM1 and NUHM2 also offer more
possibilities than the CMSSM for small cross sections for lower neutralino
masses, since they may be suppressed by scalar and pseudoscalar Higgs masses
that are larger than in the CMSSM.
