Mantle convection on Mars and Venus is likely to occur in the regime known as stagnant lid convection. We perform thermal boundary layer analyses as well as finite element simulations of stagnant lid convection with non-Newtonian viscosity (which is believed to be more appropriate for the lithosphere and upper mantle) and discuss one particular application of the results, the efficiency of heat transport on the terrestrial planets. As in the case of Newtonian viscosity, the efficiency of heat transfer in the stagnant lid regime is extremely low compared to plate tectonics: For example, in the absence of plate tectonics, the mantle temperature on Earth, which is already close to the solidus, would be about 700--1500 K higher for the present-day value of the surface heat flux. For Venus, the critical heat flux which can be removed without widespread melting is only 10--20 mW/m2. For Mars, it is 15--30 mW/m2. Therefore, there are no doubts that in the absence of mobile plates, the mantle temperature would significantly exceed solidus during planetary evolution. It is hypothesized that this could cause one, or a combination, of two possible processes: (1) differentiation of radiogenic isotopes into the crust during early planetary magmatism and (2) initiation of some kind of plate tectonics as a result of plate weakening due to melting. ¿ 1998 American Geophysical Union