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Enrico Bozzo, Chikako Ishizuka, David Ian Jones, Micaela Oertel, Constança Providencia, Laura, Tolos and Stefan Typel contributed equally to this study. The CompOSE Topical Issue is a compendium of several works on neutron star equations of state (EoS) and transport properties related to the online repository CompOSE (CompStar Online Supernovae Equations of State). [Extracted from the article]

We review the equation of state (EoS) models covering a large range of temperatures, baryon number densities and electron fractions presently available on the CompOSE database. These models are intended to be directly usable within numerical simulations of core-collapse supernovae, binary neutron star mergers and proto-neutron star evolution. We discuss their compliance with existing constraints from astrophysical observations and nuclear data. For a selection of purely nucleonic models in reasonable agreement with the above constraints, after discussing the properties of cold matter, we review thermal properties for thermodynamic conditions relevant for core-collapse supernovae and binary neutron star mergers. We find that the latter are strongly influenced by the density dependence of the nucleon effective mass. The selected bunch of models is used to investigate the EoS dependence of hot star properties, where entropy per baryon and electron fraction profiles are inspired from proto-neutron star evolution. The Γ -law analytical thermal EoS used in many simulations is found not to describe well these thermal properties of the EoS. However, it may offer a fair description of the structure of hot stars whenever thermal effects on the baryonic part are small, as shown here for proto-neutron stars starting from several seconds after bounce. [ABSTRACT FROM AUTHOR]

In this review we discuss self-consistent methods to calculate the global structure of strongly magnetised neutron stars within the general-relativistic framework. We outline why solutions in spherical symmetry cannot be applied to strongly magnetised compact stars, and elaborate on a consistent formalism to compute rotating magnetised neutron star models. We also discuss an application of the above full numerical solution for studying the influence of strong magnetic fields on the radius and crust thickness of magnetars. The above technique is also applied to construct a "universal" magnetic field profile inside the neutron star, that may be useful for studies in nuclear physics. The methodology developed here is particularly useful to interpret multi-messenger astrophysical data of strongly magnetised neutron stars. [ABSTRACT FROM AUTHOR]