  We review various theoretical approaches for the equation of state (EoS) of
dense matter, relevant for the description of core-collapse supernovae, compact
stars and compact star mergers. The emphasis is put on models that are
applicable to all of these scenarios. Such EoS models have to cover large
ranges in baryon number density, temperature and isospin asymmetry. The
characteristics of matter change dramatically within these ranges, from a
mixture of nucleons, nuclei, and electrons to uniform, strongly interacting
matter containing nucleons, and possibly other particles such as hyperons or
quarks. As the development of an EoS requires joint efforts from many
directions we consider different theoretical approaches and discuss relevant
experimental and observational constraints which provide insights for future
research. Finally, results from applications of the discussed EoS models are
summarized.
