Subjecting liquid helium to pressures of several tens of bar results in the formation of solid crystals. These are probably the most remarkable solids in nature, since the atoms can actually move between lattice sites, even at the lowest temperatures. This occurs by a process of quantum tunnelling, arising from the large zero-point motion of the atoms. A related feature of solid 4He is that dilute impurities, such as vacancies or 3He atoms, behave as quasi-particle excitations, exhibiting long-range ballistic diffusion through the crystal. It may well be that extended defects such as dislocations also exhibit quantum mobility. Some experimentalists claim that solid 4He is a supersolid at low enough temperatures, forming a quantum coherent state that is able to flow without resistance. We are investigating these unusual and fascinating phenomena, which are attracting considerable interest and controversy, with torsional oscillator and NMR techniques.