  We show how redshift-space distortions of the galaxy correlation function or
power spectrum can constrain the matter density parameter Omega_m and the
linear matter fluctuation amplitude sigma_8. We improve on previous treatments
by adopting a fully non-linear description of galaxy clustering and bias, which
allows us to break parameter degeneracies by combining large-scale and small-
scale distortions. We consider different combinations of Omega_m and sigma_8
and find parameters of the galaxy halo occupation distribution (HOD) that yield
nearly identical galaxy correlation functions in real space. We use these HOD
parameters to populate the dark matter halos of large N-body simulations, from
which we measure redshift-space distortions on small and large scales. We
include a velocity bias parameter alpha_v that allows the velocity dispersions
of satellite galaxies in halos to be systematically higher or lower than those
of dark matter. Large-scale distortions are determined by the parameter
combination beta = Omega_m^{0.6}/b_g, where b_g is the galaxy bias, in
agreement with linear theory. However, linear theory does not accurately
describe the distortions themselves on scales accessible to our simulations. We
provide fitting formulas to estimate beta from the redshift-space correlation
function or power spectrum, and we show that these formulas are significantly
more accurate than those in the existing literature. On small scales, the
``finger-of-god'' distortions at projected separations ~0.1 Mpc/h depend on
Omega_m*alpha_v^2 but are independent of sigma_8, while at intermediate
separations they depend on sigma_8 as well. One can thus use redshift-space
distortions over a wide range of scales to separately determine Omega_m,
sigma_8, and alpha_v. (Abridged)
