Plate tectonics and convection of the solid, rocky mantle are responsible for transporting heat out of Earth. However, the physics of plate tectonics is poorly understood; other planets do not exhibit it. Recent seismic evidence for convection and mixing throughout the mantle seems at odds with the chemical composition of erupted magmas requiring the presence of several chemically distinct reservoirs within the mantle. There has been rapid progress on these two problems, with the emergence of the first self-consistent models of plate tectonics and mantle convection, along with new geochemical models that may be consistent with seismic and dynamical constraints on mantle structure.