The static and dynamic properties of dilute and semidilute salt-free polyelectrolyte solutions are studied using Brownian dynamics simulations. The polymer molecules and counterions are modeled as charged bead-spring chains and charged spheres, respectively. Results are presented for the static properties, polymer rotational dynamics, self and collective diffusion coefficients, intermediate scattering functions, and dynamic structure factors of polyelectrolytes and counterions. The simulations reveal a strong dynamical coupling between counterions and polyions at low concentrations; this coupling becomes weaker as the concentration is increased. The polyion self-diffusion coefficient shows a nonmonotonic concentration dependence, but the counterion self diffusion coefficient decreases monotonically as the concentration is increased. In semidilute solutions, the collective density relaxation dynamics are slowed down at wave vectors where there is a peak in the corresponding static structure factor. This can be qualitatively, but not quantitatively understood via the concept of de Gennes narrowing. A comparison with relevant experiments and theories is also presented.