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At the elevated pressures and temperatures of the Earth's deep interior, solid rock responds to stress by slow, creeping flow. Thermal convection in the mantles of the Earth, Mars, and Venus drive geological activity on the surfaces of these planets, but one of the major open questions in earth science concerns the nature of this flow. Mantle convection is studied primarily using computational methods on the Boussinesq approximation of the conservation equations for mass, momentum, and energy. In this talk, I will discuss current ideas and models of convection in the terrestrial planets, along with some of the computational and conceptual challenges that remain.