This dissertation deals with the equation of state of hot and dense matter in compact stars, with special focus on first order phase transitions. A general classification of first order phase transitions is given and the properties of mixed phases are discussed. Aspects of nucleation and the role of local constraints are investigated. The theoretical concepts are applied to matter in the hadron-quark and the liquid-gas phase transition. For the detailed description of the liquid-gas phase transition a new nuclear statistical equilibrium model is developed. Different equation of state tables are calculated and the composition and thermodynamic properties of supernova matter are analyzed. As a first application numerical simulations of core-collapse supernovae are presented. For the hadron-quark phase transition two possible scenarios are studied in more detail, the appearance of a new mixed phase in a protoneutron star and the consequences of the hadron-quark transition in a core-collapse supernovae. Simulations of the latter show that the appearance of quark matter has clear observable signatures and can even lead to the generation of an explosion.