The mechanisms of crystallization and melting of polymers are richly different from those of small molecular systems. As examples, the path of crystallization is different from that of melting and these two are not reversible, as reflected in different crystallization and melting temperatures for the same polymer, and polymer recrystallizes while melting. Another example is the memory of a thoroughly homogenized melt undergoing crystallization that it was a crystal before forming the melt. The primary origin of the various enigmatic behaviors of polymer crystallization/melting phenomenon lies in the conformational entropy and entanglements resulting from intertwined chain connectivity. Based on our statistical mechanics theory and Langevin dynamics simulations, we will present recent conceptual advances in the fundamental understanding of polymer crystallization, melting, and melt memory.