  We calculate the number of damped Lyman-alpha absorbers expected in various
popular cosmological models as a function of redshift and compare our
predictions with observed abundances. The Press-Schechter formalism is used to
obtain the distribution of halos with circular velocity in different
cosmologies, and we calibrate the relation between circular velocity and
absorption cross-section using detailed gas dynamical simulations of a
``standard'' cold dark matter (CDM) model. Because of this calibration, our
approach makes more realistic assumptions about the absorption properties of
collapsed objects than previous, analytic calculations of the damped
Lyman-alpha abundance. CDM models with Omega_0=1, H_0=50, baryon density
Omega_b=0.05, and scale-invariant primeval fluctuations reproduce the observed
incidence and redshift evolution of damped Lyman-alpha absorption to within
observational uncertainty, for both COBE normalization (sigma_8=1.2) and a
lower normalization (sigma_8=0.7) that better matches the observed cluster
abundance at z=0. A tilted (n=0.8, sigma_8=0.7) CDM model tends to underproduce
absorption, especially at z=4. With COBE normalization, a CDM model with
Omega_0=0.4, Omega_{Lambda}=0.6 gives an acceptable fit to the observed
absorption; an open CDM model is marginally acceptable if Omega_0 is at least
0.4 and strongly inconsistent with the z=4 data if Omega_0=0.3. Mixed dark
matter models tend not to produce sufficient absorption, being roughly
comparable to tilted CDM models if Omega_{nu} = 0.2 and failing drastically if
Omega_{nu} = 0.3.
