  Dark-matter halos are the scaffolding around which galaxies and clusters are
built. They form when the gravitational instability of primordial density
fluctuations causes regions which are denser than average to slow their cosmic
expansion, recollapse, and virialize. Understanding the equilibrium structure
of these halos is thus a prerequisite for understanding galaxy and cluster
formation. Numerical N-body simulations of structure formation from
Gaussian-random-noise initial conditions in the CDM universe find a universal
internal structure for halos. Objects as different in size and mass as dwarf
spheroidal galaxies and galaxy clusters are predicted to have halos with the
same basic structure when properly rescaled, independent of halo mass, of the
shape of the power spectrum of primordial density fluctuations, and of the
cosmological background parameters. This remarkable universality is a
fundamental prediction of the CDM model, but our knowledge is limited to the
``empirical'' N-body simulation results, with little analytical understanding.
We summarize here our attempts to fill this gap, in an effort to derive and
give physical insight to the numerical results and extend them beyond the range
of numerical simulation: (1) Simulated halos which form from highly simplified
initial conditions involving gravitational instability in a cosmological
pancake show that many of the universal properties of CDM halos are generic to
cosmological gravitational collapse and do not require Gaussian-random-noise
density fluctuations or hierarchical clustering. (2) A fluid approximation
derived from the Boltzmann equation yields an analytical theory of halo
dynamics which can explain many of the N-body results ... (abridged)
