  The geometric theory of vortex tunnelling in superfluid liquids is developed.
Geometry rules the tunnelling process in the approximation of an incompressible
superfluid, which yields the identity of phase and configuration space in the
vortex collective co-ordinate. To exemplify the implications of this approach
to tunnelling, we solve explicitly for the two-dimensional motion of a point
vortex in the presence of an ellipse, showing that the hydrodynamic collective
co-ordinate description limits the constant energy paths allowed for the vortex
in configuration space. We outline the experimental procedure used in helium II
to observe tunnelling events, and compare the conclusions we draw to the
experimental results obtained so far. Tunnelling in Fermi superfluids is
discussed, where it is assumed that the low energy quasiparticle excitations
localised in the vortex core govern the vortex dynamical equations. The
tunnelling process can be dominated by Hall or dissipative terms, respectively
be under the influence of both, with a possible realization of this last
intermediate case in unconventional, high-temperature superconductors.
