We review the field of solid-phase epitaxy (SPE), a crystallization process during which atoms undergo bonding rearrangements that allow them to transfer from a metastable amorphous phase to a crystalline phase using the substrate as a template. This process is commonly used to activate an implanted dopant profile on a low thermal budget, often to concentrations much greater than the dopant’s solid solubility limit. Two model systems with great technological importance are used to discuss SPE: silicon and germanium. The well-characterized dependence of SPE on various experimental parameters are considered in detail as well as kinetic models that predict the SPE velocity associated with the crystal-amorphous interface motion. Atomistic models are also reviewed and are shown to provide great insight into the possible mechanisms responsible for the process. Observations of technological interest to current and future semiconductor device fabrication are also discussed, including dopant activation and defect formation.