  We discuss the statistics of tunnelling rates in the presence of chaotic
classical dynamics. This applies to resonance widths in chaotic metastable
wells and to tunnelling splittings in chaotic symmetric double wells. The
theory is based on using the properties of a semiclassical tunnelling operator
together with random matrix theory arguments about wave function overlaps. The
resulting distribution depends on the stability of a specific tunnelling orbit
and is therefore not universal. However it does reduce to the universal
Porter-Thomas form as the orbit becomes very unstable. For some choices of
system parameters there are systematic deviations which we explain in terms of
scarring of certain real periodic orbits. The theory is tested in a model
symmetric double well problem and possible experimental realisations are
discussed.
