In this chapter, we investigate the structure and composition of hot neutron star matter and proto-neutron stars. Such objects are made of baryonic matter that is several times denser than atomic nuclei and tens of thousands times hotter than the matter in the core of our Sun. The relativistic finite-temperature Green function formalism is used to formulate the expressions that determine the properties of such matter in the framework of the density-dependent mean field approach. Three different sets of nuclear parametrizations are used to solve the many-body equations and to determine the models for the equation of state of ultra-hot and dense stellar matter. The meson-baryon coupling scheme and the role of the ∆( 1232) baryon in proto-neutron star matter are investigated in great detail. In addition, using the non-local three-flavor Nambu-Jona-Lasinio model to describe quark matter, the hadron-quark composition of dense baryonic matter at zero temperature is discussed briefly. General relativistic models of non-rotating as well as rotating proto-neutron stars are presented in part two of our study.