We review the role of the surface tension of quark matter droplets in astrophysical conditions, focusing specifically on the thermodynamic conditions prevailing in cold neutron stars (NSs), in hot lepton rich proto NSs, and in early universe conditions. We analyze quark matter in chemical equilibrium under weak interactions, which is relevant for understanding the internal composition of hybrid stars, as well as just deconfined quark matter out of chemical equilibrium, which is the relevant thermodynamic state for describing the nucleation process of quark matter in NSs. We explore the role of temperature, density, trapped neutrinos, droplet size and magnetic fields within the multiple reflection expansion formalism (MRE). Quark matter is described within the frame of different effective models: the MIT bag model and the $SU(3)_f$ Nambu-Jona-Lasinio model (NJL), including color superconductivity, neutrino trapping and magnetic fields. We also analyze the deconfinement transition at vanishing chemical potential and finite temperature including the Polyakov loop. We explore some astrophysical consequences of our results.
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