The QCD equation of state from the lattice

The equation of state of QCD at finite temperatures and baryon densities has a wide range of applications in many fields of modern particle and nuclear physics. It is the main ingredient to describe the dynamics of experimental heavy ion collisions, the expansion of the early universe in the standard model era and the interior of compact stars. On most scales of interest, QCD is strongly coupled and not amenable to perturbative investigations. Over the past decade, first principles calculations using lattice QCD have reached maturity, in the sense that for particular discretisation schemes simulations at the physical point have become possible, finite temperature results near the continuum limit are available and systematic errors begin to be controlled. This review summarises the current theoretical and numerical state of the art based on staggered and Wilson fermions.