Hydrogen migration in solid-state crystallized and low-pressure chemical-vapor-deposited (LPCVD) polycrystalline silicon (poly-Si) was investigated by deuterium diffusion experiments. The concentration profiles of deuterium, introduced into the poly-Si samples either from a remote D plasma or from a deuterated amorphous-silicon layer, were measured as a function of time and temperature. At high deuterium concentrations the diffusion was dispersive depending on exposure time. The dispersion is consistent with multiple trapping within a distribution of hopping barriers. The data can be explained by a two-level model used to explain diffusion in hydrogenated amorphous silicon. The energy difference between the transport level and the deuterium chemical potential was found to be about 1.2--1.3 eV. The shallow levels for hydrogen trapping are about 0.5 eV below the transport level, while the deep levels are about 1.5--1.7 eV below. The hydrogen chemical potential ${\mathrm{\ensuremath{\mu}}}_{\mathrm{H}}$ decreases as the temperature increases. At lower concentrations, ${\mathrm{\ensuremath{\mu}}}_{\mathrm{H}}$ was found to depend markedly on the method used to prepare the poly-Si, a result due in part to the dependence of crystallite size on the deposition process. Clear evidence for deuterium deep traps was found only in the solid-state crystallized material. The LPCVD-grown poly-Si, with columnar grains extending through the film thickness, displayed little evidence of deep trapping, and exhibited enhanced D diffusion. Many concentration profiles in the columnar LPCVD material indicated complex diffusion behavior, perhaps reflecting spatial variations of trap densities, complex formation, and/or multiple transport paths. Many aspects of the diffusion in poly-Si are consistent with diffusion data obtained in amorphous silicon. \textcopyright{} 1996 The American Physical Society.