In this paper, we examine collective and self-diffusion properties of dispersions of spherically shaped colloidal particles at intermediate and long times. Our analysis is based on a fully self-consistent (rescaled) mode coupling theory (MCT) adjusted to describe the overdamped dynamics in concentrated suspensions of neutral and charged colloidal particles. The dynamical quantities studied in dependence on various experimentally controllable system parameters are the particle mean-squared displacement, long-time collective and self-diffusion coefficients, dynamic structure factors, nonexponentiallity factors and collective and self-memory functions. The results of our theoretical treatment are compared with Brownian dynamics computer simulation data, experiment and other existing theories. It is shown that the rescaled MCT can be successfully applied to a wide range of dynamical properties. Our calculations reveal in particular an exponential long-time mode of the dynamic structure factor for a limited range of wave numbers and at sufficiently high concentrations. A dynamic scaling behavior of the dynamic structure factor and self-intermediate scattering function is predicted for the important case of salt-free charge-stabilized suspensions. As a consequence of the dynamic scaling, the static freezing criterion for colloids by Hansen and Verlet [Phys. Rev. 184, 151 (1969)] is shown to be equivalent with the dynamic criterion by Lowen [Phys. Rev. Lett. 70, 1557 (1993)] related to long-time self-diffusion.