The theory of the tunnelling of waves through a barrier in which the square of the effective refractive index is zero at one boundary and infinite at or near the other is studied. An infinity of the refractive index is called a resonance and so we speak of resonance tunnelling. The sum of the powers in the reflected and transmitted waves is less than the power in the incident wave even in a loss free system where there is no mechanism for the absorption of energy. A formal proof is given that there must be such a disappearance of energy, associated with the solution of the governing equations that is singular at the resonance. The problem of what has happened to the lost energy is discussed. Some previous treatments dealt only with normally incident waves, but this is a degenerate case. The theory is extended to include oblique incidence and some new features are revealed. Some specific examples are worked out as illustrations.