In the work presented in this dissertation, extensive molecular dynamics (MD) simulations have been performed to investigate various physical problems related to the solid-liquid transitions over a wide range of supercooling and superheating temperatures in model Lennard-Jones systems. The major focus of this work is to investigate the thermodynamics, kinetics, and underlying mechanisms of these problems. There are five topics in this work: (1) The classical nucleation theory (CNT) was tested for both liquid supercooling and solid superheating via different solid-liquid coexistence models. It is found that the CNT is valid for liquid supercooling but invalid for solid superheating. The arising elastic energy plays a significant role in affecting the liquid nucleation in a superheated solid. A new nucleation theory was proposed for describing the internal liquid nucleation of solid superheating. (2) Based on CNT, a new and accurate method was developed for calculating the crystal-melt interfacial free energy and its anisotropy. Our result is very close to Turnbulls experimental results. (3) The face, temperature, and size dependences of the crystallization rate were investigated in this work. The results show that the crystallization rate decreases substantially with the increasing system size. Different from the conventional models, a new model is developed to describe these dependences. (4) Melting from internal nanovoids was investigated in this work. It is found that the mechanism of void melting is quite different from bulk melting and nanoparticle melting. There are four different stages and three local melting temperatures in void melting. The mechanism of the complex melting sequence is systematically explained. (5) The homogenous melting at the upper limit of superheating was investigated in this work. For the first time, the ring diffusion is found to take place in superheated crystals and causes the spontaneous melting. The prevailing instability theories are unsuitable to describe this type of melting. The mechanism of the diffusion-loop mediated melting is carefully discussed in this work.