The correlation function of dark matter particles can be written as the sum of two terms---one which accounts for nonlinear evolution and dominates on small scales, and another which is essentially the term from linear theory and dominates on large scales. Accurate models of the number and spatial distribution of haloes and halo density profiles allow one to describe how these terms evolve accurately. This decomposition is useful when interpretting weak-lensing results. I show that it also provides simple and accurate models of how the first and second moments of the pairwise velocity distribution depend on scale. Velocities of particles can also be decomposed into linear and nonlinear parts. I show that doing so provides a simple model of how the nonlinearly evolved power-spectrum of dark matter velocities today differs from that predicted by linear theory. This also provides a simple model for the single particle velocity distribution function. When combined with the pairwise velocity results above, this allows a simple description of redshift-space distortions over the entire range of linear to highly nonlinear regimes. Finally, I discuss how these models for the dark matter distribution can be extended to provide estimates of the correlation function and peculiar velocities of galaxies.