The anisotropic magnetoresistance (AMR) at low temperatures is theoretically studied for low-dimensional NiFe-based systems in various geometries by solving the Boltzmann transport equation. The AMR is treated by introducing spin-dependent anisotropic mean free paths, making use of anisotropic-scattering parameters that are extracted from experimental spin-resolved resistivity data for bulk dilute NiFe alloys. A first set of calculations comprises the AMR in NiFe thin films and cylindrical wires, as a function of the layer thickness and wire diameter, respectively. For the thin film case we have considered rotation of the magnetization vector within the film plane as well as out of the film plane. For the latter the highest AMR ratio is found, which even slightly exceeds the bulk value. For wires the dependence of the AMR on the dimensions is qualitatively different as compared to the film case due to the relatively enhanced importance of boundary scattering. Finally, the validity of a description of the combined effect of AMR and the giant magnetoresistance in terms of a simple summation of the two effects is studied by performing model calculations for NiFe/Cu/NiFe trilayers.