  These notes were written for the mini-course "Extrema of log-correlated
random variables: Principles and Examples" at the Introductory School held in
January 2015 at the Centre International de Rencontres Math\'ematiques in
Marseille. There have been many advances in the understanding of the high
values of log-correlated random fields from the physics and mathematics
perspectives in recent years. These fields admit correlations that decay
approximately like the logarithm of the inverse of the distance between index
points. Examples include branching random walks and the two-dimensional
Gaussian free field. In this paper, we review the properties of such fields and
survey the progress in describing the statistics of their extremes. The
branching random walk is used as a guiding example to prove the correct leading
and subleading order of the maximum following the multiscale refinement of the
second moment method of Kistler. The approach sheds light on a conjecture of
Fyodorov, Hiary & Keating on the maximum of the Riemann zeta function on an
interval of the critical line and of the characteristic polynomial of random
unitary matrices.
