Hydrogen and helium are the most abundant elements in the Universe. They are also, in principle, the most simple. Nonetheless, they display remarkable properties under extreme conditions of pressure and temperature that have fascinated theoreticians and experimentalists for over a century. Advances in computational methods have made it possible to elucidate ever more of their properties. Some of these methods that have been applied in recent years, in particular, those that perform simulations directly from the physical picture of electrons and ions, such as density functional theory and quantum Monte Carlo are reviewed. The predictions from such methods as applied to the phase diagram of hydrogen, with particular focus on the solid phases and the liquid-liquid transition are discussed. The predictions of ordered quantum states, including the possibilities of a low- or zero-temperature quantum fluid and high-temperature superconductivity are also considered. Finally, pure helium and hydrogen-helium mixtures, the latter which has particular relevance to planetary physics, are discussed.
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