Graphical abstract:Abstract: Supercooled liquids give rise, by homogeneous nucleation, to solid superclusters acting as building blocks of glass, ultrastable glass, and glacial glass phases before being crystallized. Liquid-to-liquid phase transitions begin to be observed above the melting temperature Tm as well as critical undercooling depending on critical overheating T/Tm. Solid nuclei exist above Tm and melt by homogeneous nucleation of liquid instead of surface melting. The Gibbs free energy change predicted by the classical nucleation equation is completed by an additional enthalpy which stabilize these solid entities during undercooling. A two-liquid model, using this renewed equation, predicts the new homogeneous nucleation temperatures inducing first-order transitions, and the enthalpy and entropy of new liquid and glass phases. These calculations are successfully applied to ethylbenzene, triphenyl phosphite, d-mannitol, n-butanol, Zr41.2Ti13.8Cu12.5Ni10Be22.5, Ti34Zr11Cu47Ni8, and Co81.5B18.5. A critical supercooling and overheating rate T/Tm = 0.198 of liquid elements is predicted in agreement with experiments on Sn droplets.