Various melting-related phenomena (like surface melting, size dependence of melting temperature, melting of few nm-size particles and overheating at a very fast heating rate) are of great fundamental and applied interest, although the corresponding theory is still lacking. Here we develop an advanced phase-field theory of melting coupled to mechanics, which resolves numerous existing contradictions and allowed us to reveal exciting features of melting problems. The necessity of introducing an unexpected concept, namely, coherent solid-melt interface with uniaxial transformation strain, is demonstrated. A crossover in temperature dependence of interface energy for radii below 20 nm is found. Surface-induced premelting and barrierless melt nucleation for nanoparticles down to 1 nm radius is studied, and the importance of advanced mechanics is demonstrated. Our model describes well experimental data on the width of the molten layer versus temperature for the Al plane surface and on melting temperature versus particle radius.