The influence of drag on the motion of gravity currents over rigid horizontal surfaces is considered analytically using a Chézy model of boundary shear stress. Although the initial motion is governed by a balance between the buoyancy forces and fluid inertia, drag gradually influences the flow. The length and time scales at which these effects become significant are identified. A perturbation series, valid at early times, is constructed to analyse the changes to the velocity and height of the evolving current due to drag. At much later times, a new class of similarity solutions is developed to model the motion which is now governed by a balance between buoyancy and drag. The transition in the dominant forces which govern the dynamics of the flow is examined by numerically integrating the equations of motion for flows generated by a constant flux of relatively dense fluid. The numerical results confirm both the perturbation solution, valid at early times, and the new similarity solution valid at late times. The transition between the two may involve the formation of a discontinuity (bore). Finally particle-driven currents, which exhibit different dynamical behaviour due to the progressive reduction of their density arising from particle sedimentation, are investigated.