The first detection of gravitational waves from a merging neutron star
binary system and the accompanying observations of electromagnetic
counterparts in 2017 demonstrated the enormous potential of
multi-messenger astronomy for understanding the properties of
ultra-dense (and hot) matter. Neutron stars --relict of the
gravitational collapse and subsequent supernova explosion of a massive
star at the end of his life-- comprise the highest densities of matter
that can stably exist in the Universe. During this talk, I will
discuss some details about the complex microphysics involved and the
possibilities for insights on matter under these extreme conditions
from future detections of neutron star related astrophysical
events. Among others, results will be shown illustrating the capacity
of multi-messenger observations to give us hints about a possible
phase transition in neutron star matter.