We address the question of how convective processes control the thicknesses of oceanic and continental lithospheres. The numerical convection model involves a Newtonian rheology which depends on temperature and pressure. A repeated plate tectonic cycle is modeled by imposing a time-dependent surface velocity. One part of the surface, representing a continent, never subducts. The asymptotic equilibrium thickness of the lithosphere varies with the viscosity at the base of the lithosphere, but is not directly sensitive to the pressure dependence of the viscosity law and to the plate velocity. For small activation volumes, and average upper mantle viscosities deduced from postglacial rebound, the equilibrium plate thickness is more than 400 km (regimes 1 and 2). The equilibrium thickness of the oceanic lithosphere (around 100 km) implies that the viscosity in the asthenosphere is less than 7¿1018 Pa s. Only models with strongly pressure-dependent viscosity laws (activation volumes greater than 9¿10-6 m3/mol) are able to reconcile this value with the average upper mantle viscosity (5¿1020 Pa s). For these models, there are two lithospheric thicknesses such that the heat supplied by convection at the base of the lithosphere equals the surface conductive heat flow (regime 3). They could be that of an aged oceanic lithosphere and that of a shield lithospheric root. They indeed appear as points of preferred thickness in our numerical models. However, convection triggered by the lateral density jumps at the boundaries between the root and the thinner lithosphere slowly destabilizes the thick lithosphere. A plausible degree of chemical buoyancy in a depleted lithospheric root does not prevent convective erosion. In our simulations, long-term stability of a cratonic lithospheric root is best achieved when its material is both buoyant and more viscous than the surrounding mantle. Extensive devolatilization of the refractory rocks forming the root is invoked to explain this viscosity increase.¿ 1997 American Geophysical Union