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We review the role of the surface tension of quark matter droplets inastrophysical conditions, focusing specifically on the thermodynamic conditionsprevailing in cold neutron stars (NSs), in hot lepton rich proto NSs, and inearly universe conditions. We analyze quark matter in chemical equilibriumunder weak interactions, which is relevant for understanding the internalcomposition of hybrid stars, as well as just deconfined quark matter out ofchemical equilibrium, which is the relevant thermodynamic state for describingthe nucleation process of quark matter in NSs. We explore the role oftemperature, density, trapped neutrinos, droplet size and magnetic fieldswithin the multiple reflection expansion formalism (MRE). Quark matter isdescribed within the frame of different effective models: the MIT bag model andthe $SU(3)_f$ Nambu-Jona-Lasinio model (NJL), including colorsuperconductivity, neutrino trapping and magnetic fields. We also analyze thedeconfinement transition at vanishing chemical potential and finite temperatureincluding the Polyakov loop. We explore some astrophysical consequences of ourresults.

Surface tension of quark droplets in compact stars and in the early universe