  We consider the minimal supersymmetric extension of the Standard Model (MSSM)
with varying amounts of non-universality in the soft supersymmetry-breaking
contributions to the Higgs scalar masses. In addition to the constrained MSSM
(CMSSM) in which these are universal with the soft supersymmetry-breaking
contributions to the squark and slepton masses at the input GUT scale, we
consider scenarios in which both the Higgs scalar masses are non-universal by
the same amount (NUHM1), and scenarios in which they are independently
non-universal (NUHM2). We show how the NUHM1 scenarios generalize the (m_{1/2},
m_0) planes of the CMSSM by allowing either mu or m_A to take different (fixed)
values and we also show how the NUHM1 scenarios are embedded as special cases
of the more general NUHM2 scenarios. Generalizing from the CMSSM, we find
regions of the NUHM1 parameter space that are excluded because the LSP is a
selectron. We also find new regions where the neutralino relic density falls
within the range preferred by astrophysical and cosmological measurements,
thanks to rapid annihilation through direct-channel Higgs poles, or
coannihilation with selectrons, or because the LSP composition crosses over
from being mainly bino to mainly Higgsino. Generalizing further to the NUHM2,
we find regions of its parameter space where a sneutrino is the LSP, and others
where neutralino coannihilation with sneutrinos is important for the relic
density. In both the NUHM1 and the NUHM2, there are slivers of parameter space
where the LHC has fewer prospects for discovering sparticles than in the CMSSM,
because either m_{1/2} and/or m_0 may be considerably larger than in the CMSSM.
