Semiconductor heterostructures with modulated composition and/or doping enable passivation of interfaces and the generation of devices with diverse functions. In this regard, the control of interfaces in nanoscale building blocks with high surface area will be increasingly important in the assembly of electronic and photonic devices. Core-shell heterostructures formed by the growth of crystalline overlayers on nanocrystals offer enhanced emission efficiency, important for various applications. Axial heterostructures have also been formed by a one-dimensional modulation of nanowire composition and doping. However, modulation of the radial composition and doping in nanowire structures has received much less attention than planar and nanocrystal systems. Here we synthesize silicon and germanium core-shell and multishell nanowire heterostructures using a chemical vapour deposition method applicable to a variety of nanoscale materials. Our investigations of the growth of boron-doped silicon shells on intrinsic silicon and silicon-silicon oxide core-shell nanowires indicate that homoepitaxy can be achieved at relatively low temperatures on clean silicon. We also demonstrate the possibility of heteroepitaxial growth of crystalline germanium-silicon and silicon-germanium core-shell structures, in which band-offsets drive hole injection into either germanium core or shell regions. Our synthesis of core-multishell structures, including a high-performance coaxially gated field-effect transistor, indicates the general potential of radial heterostructure growth for the development of nanowire-based devices.