The Earth's mantle is almost entirely solid, but on geological timescales it convects vigorously, the well-known surface expression of this being plate tectonics. At depths up to ~100 km beneath plate-tectonic boundaries (mid-ocean ridges and subduction zones), and beneath ocean islands such as Hawaii, the mantle melts, and that melt rises to the surface to feed volcanism and form new crust. Such magmatism plays a key role in the chemical evolution and dynamics of our planet. Although the basic thermodynamics of melt generation in these settings is well understood, how the melt is transported to the surface is not, despite several decades of work on the problem. Furthermore, recent observational evidence suggests that mantle melting is not restricted to the near surface (top 100 km): it may occur within the mantle transition zone (410-660 km depth) and above the core-mantle boundary (2900 km). For these deeper instances of melting, an understanding of the dynamical and thermochemical characteristics is currently lacking.