A comprehensive introduction to the theoretical foundations of quantum tunnelling, stressing the basic physics underlying the applications. The topics addressed include exponential and non-exponential decay processes and the application of scattering theory to tunnelling problems. In addition to the Schrodinger equation approach, the path integral, Heisenberg’s equations and the phase space method are all used to study the motion of a particle under the barrier. Extensions to the multidimensional cases and tunnelling of particles with internal degrees of freedom are also considered. Furthermore, advances concerning time delay and tunnelling times and some of the problems associated with their measurement are also discussed. Finally, some examples of tunnelling in atomic, molecular, nuclear and condensed matter physics are presented.