Phase Conversions in Neutron Stars: Implications for Stellar Stability and Gravitational Wave Astrophysics

We review the properties of hybrid stars with a quark matter core and hadronic mantle, focusing on role key micro-physical such as quark/hadron surface curvature tensions conversion speed at interface between both phases. summarize results works that have determined from microscopic calculations. If these quantities are large enough, mixed phases energetically suppressed would be separated mantle by sharp interface. is slow, new class dynamically stable objects possible. Densities tens times larger than nuclear saturation density can attained center objects. discuss possible formation mechanisms for smoking guns their observational identification.