What do we understand of the properties of the dense ionised matter found in the interiors of low-mass stars and giant planets? The 147th IAU Colloquium gathered together international experts to address this question, and their answers are provided in this opportune proceedings. In this volume, reviews by world experts in plasma and dense matter physics and in stellar astrophysics cover everything from the cooling theory of white dwarfs and their accretion-induced collapse, through to the internal structure of low-mass stars and giant planets. They cover a wide range of topics related to the equation of state in dense matter. Together these articles provide an essential review of the most recent achievements in the field and give directions for future research.