This is the first paper of a series of reports concerning extended-chain crystals of flexible, linear high polymers. The general conditions for crystal growth are discussed. Polymer crystallization is described as a two-step process: nucleation of each crystalizing molecule to a folded-chain conformation, followed by an increase in fold length in a solid-state reorganization step. This reorganization step is enhanced in the case of polyethylene by crystallization at high temperature under elevated pressure. Mechanical deformation during crystallization is also able to produce extended-chain crystals. The most promising method, however, is crystallization during polymerization. Previous work on crystallization of polyethylene under elevated pressure is critically reviewed.