  In statistical physics, the XY model in two dimensions provides the
paradigmatic example of phase transitions mediated by topological defects
(vortices). Over the years, a variety of analytical and numerical methods have
been deployed in an attempt to fully understand the nature of its transition,
which is of the Berezinskii-Kosterlitz-Thouless type. These met with only
limited success until it was realized that subtle effects (logarithmic
corrections) that modify leading behaviour must be taken into account. This
realization prompted renewed activity in the field and significant progress has
been made. This paper contains a review of the importance of such subtleties,
the role played by vortices and of recent and current research in this area.
Directions for desirable future research endeavours are outlined.
