The difficulties with performing accurate experiments under the extreme pressures and temperatures of the Earth's mantle and core has ensured that computational mineral physics is a fundamental tool for understanding the properties the Earth's interior.  Essentially, computational mineral physics uses the interactions between the atoms and electrons to obtain thermodynamic and physical properties of minerals, melts, and fluids. The growth of large parallel computers has meant that we are now using "ab initio molecular dynamics" to calculate accurately the properties of minerals and melts at the high temperatures and pressures of planetary interiors.  Previously, these accurate ab initio techniques were only used at zero temperature and more approximate methods had to be used for high temperature properties. In this talk I will briefly outline the computational methods and attempt to provide an appreciation  of the approximations and limitations.  I will also present some of the successes of these methods and some recent highlights, such as calculations on the high-temperature and high-pressure elastic constants of mantle minerals and the effect of alumina on the physical properties of perovskite.  I will also briefly describe a new method for calculating Gibbs free energies from ab initio methods, and an application to the light element in the core.