A quantum computer will have computational power beyond that of conventional computers, which can be exploited for solving important and complex problems, such as predicting the conformations of large biological molecules. Materials play a major role in this emerging technology, as they can enable sophisticated operations, such as control over single degrees of freedom and their quantum states, as well as preservation and coherent transfer of these states between distant nodes. Here we assess the potential of semiconductor nanowires grown from the bottom-up as a materials platform for a quantum computer. We review recent experiments in which small bandgap nanowires are used to manipulate single spins in quantum dots and experiments on Majorana fermions, which are quasiparticles relevant for topological quantum computing.