Highly resolved three-dimensional and two-dimensional simulations of gravity currents in planar and cylindrical configurations are presented. The volume of release of the heavy fluid is varied and the different phases of spreading, namely acceleration, slumping, inertial and viscous phases, are studied. The incompressible Navier–Stokes equations are solved assuming that the Boussinesq approximation is valid for small density difference. The simulations are performed for three different Reynolds numbers (Re): 895, 3450 and 8950 (this particular choice corresponds to values of Grashof number: 105, 1.5 × 106 and 107, respectively). Following their sudden release, the gravity currents are observed to go through an acceleration phase in which the maximum front velocity is reached. As the interface of the current rolls up, the front velocity slightly decreases from the maximum and levels off to a nearly constant value. At higher Re, three-dimensional disturbances grow rapidly and the currents become strongly turbulent. In contrast, in two-dimensional simulations, the rolled-up vortices remain coherent and very strong. Depending on the initial Re of the flow and on the size of the release, the current may transition from the slumping to the inertial phase, or directly to the viscous phase without an inertial phase. New criteria for the critical Re are introduced for the development of the inertial phase. Once the flow transitions to the inertial or viscous phase, it becomes fully three-dimensional. During these phases of spreading, two-dimensional approximations underpredict the front location and velocity. The enhanced vortex coherence of the two-dimensional simulations leads to strong vortex interaction and results in spurious strong time variations of the front velocity. The structure and dynamics of the three-dimensional currents are in good agreement with previously reported numerical and experimental observations.