The authors have measured the magnetoresistance of narrow GaAs/AlGaAs heterojunction wires at temperatures down to 0.04 K and in magnetic fields of up to 1.2 T. The analysis of the low-field magnetoresistance is consistent with a mobility-dependent saturation of the thermal diffusion and phase-breaking lengths. At higher fields, one-dimensional subband depopulation and Shubnikov-de Haas oscillations are present. They discuss how studies of the temperature and magnetic field dependence of these various effects provide important information on the possible nature of electronic transport in the wires.