  We present the Fr\"ohlich-Spencer proof of the
Berezinskii-Kosterlitz-Thouless transition. Our treatment includes the proof of
delocalization for the integer-valued discrete Gaussian free field at high
temperature and the proof of existence of a phase with power-law decay of
correlations in the plane rotator model with Villain interaction, both in two
dimensions. The treatment differs from the original in various technical points
and we hope it will be of benefit to the community.
