Dislocation velocities and mobilities are studied by Molecular Dynamics simulations for edge and screw dislocations in pure aluminum and nickel, and edge dislocations in Al-2.5%Mg and Al-5.0%Mg random substitutional alloys using EAM potentials. In the pure materials, the velocities of all dislocations are close to linear with the ratio of (applied stress)/(temperature) at low velocities, consistent with phonon drag models and quantitative agreement with experiment is obtained for the mobility in Al. At higher velocities, different behavior is observed. The edge dislocation velocity remains dependent solely on (applied stress)/(temperature) up to approximately 1.0 MPa/K, and approaches a plateau velocity that is lower than the smallest "forbidden" speed predicted by continuum models. In contrast, above a velocity around half of the smallest continuum wave speed, the screw dislocation damping has a contribution dependent solely on stress with a functional form close to that predicted by a radiation damping model of Eshelby. At the highest applied stresses, there are several regimes of nearly constant (transonic or supersonic) velocity separated by velocity gaps in the vicinity of forbidden velocities; various modes of dislocation disintegration and destabilization were also encountered in this regime. In the alloy systems, there is a temperature- and concentration-dependent pinning regime where the velocity drops sharply below the pure metal velocity. Above the pinning regime but at moderate stresses, the velocity is again linear in (applied stress)/(temperature) but with a lower mobility than in the pure metal.