  This work proves that the fluctuations of the cover time of simple random
walk in the discrete torus of dimension at least three with large side-length
are governed by the Gumbel extreme value distribution. This result was
conjectured for example in the book by Aldous & Fill. We also derive some
corollaries which qualitatively describe "how" covering happens. In addition,
we develop a new and stronger coupling of the model of random interlacements,
introduced by Sznitman, and random walk in the torus. This coupling is used to
prove the cover time result and is also of independent interest.
