Continental cratons have not experienced major tectonic disruptions over a timescale of 109 years. The thickness of cratonic lithosphere also appears to have changed little over this timescale. These observations are often attributed to the presence of chemically buoyant and/or highly viscous subcratonic roots. Simple physical scaling relationships are developed to explore the buoyancy and/or viscosity conditions required to stabilize such roots against large-scale deformation and rapid remixing into the mantle. The scalings are tested using idealized numerical simulations with good general agreement. Applied to Earth, the scalings suggest that (1) buoyancy alone is unlikely to stabilize cratonic roots and (2) if root viscosity is to provide stability into the Archean, then roots must be 103 times as viscous as the mantle. Based on available experimental data, root dehydration cannot account for the required viscosity increase. Temperature-dependent viscosity can stabilize roots, but it does so at the expense of stagnating the entire mantle lithosphere, i.e., at the expense of sacrificing plate tectonics. This suggests that the plastic yielding properties of rocks at low temperatures will need to be more directly accounted for in future experiments exploring root stability. ¿ 1999 American Geophysical Union