Silicon transistors are expected to have new gate architectures, channel materials and switching mechanisms in ten years' time. The trend in transistor scaling has already led to a change in gate structure from two dimensions to three, used in fin field-effect transistors, to avoid problems inherent in miniaturization such as high off-state leakage current and the short-channel effect. At present, planar and fin architectures using III-V materials, specifically InGaAs, are being explored as alternative fast channels on silicon because of their high electron mobility and high-quality interface with gate dielectrics. The idea of surrounding-gate transistors, in which the gate is wrapped around a nanowire channel to provide the best possible electrostatic gate control, using InGaAs channels on silicon, however, has been less well investigated because of difficulties in integrating free-standing InGaAs nanostructures on silicon. Here we report the position-controlled growth of vertical InGaAs nanowires on silicon without any buffering technique and demonstrate surrounding-gate transistors using InGaAs nanowires and InGaAs/InP/InAlAs/InGaAs core-multishell nanowires as channels. Surrounding-gate transistors using core-multishell nanowire channels with a six-sided, high-electron-mobility transistor structure greatly enhance the on-state current and transconductance while keeping good gate controllability. These devices provide a route to making vertically oriented transistors for the next generation of field-effect transistors and may be useful as building blocks for wireless networks on silicon platforms.