  Vector light beams, characterised by a spatially varying polarisation, can
exhibit localised structures reminiscent of the Skyrmions familiar from the
study of magnetic media. We present a theory of such Skyrmions within paraxial
optics, exploiting mathematical analogies with the study of superfluids,
especially the A phase of superfluid $\textrm{He}^3$. The key feature is the
Skyrmion field which, together with the underlying Skyrmion vector potential,
determines the properties of the Skyrmions and, more generally, the
polarisation structure of every paraxial vector beam. In addition to structures
with integer Skyrmion number we find polarisation patterns with non-integer
Skyrmion number; these seem to have no analogue in other fields of physics.
