Existing models of electrical contacts are often inapplicable at the nanoscale because there are significant differences between nanostructures and bulk materials arising from unique geometries and electrostatics. In this Review, we discuss the physics and materials science of electrical contacts to carbon nanotubes, semiconductor nanowires and graphene, and outline the main research and development challenges in the field. We also include a case study of gold contacts to germanium nanowires to illustrate these concepts.