  We review recent progress relating to the extreme value statistics of the
characteristic polynomials of random matrices associated with the classical
compact groups, and of the Riemann zeta-function and other $L$-functions, in
the context of the general theory of logarithmically-correlated Gaussian
fields. In particular, we focus on developments related to the conjectures of
Fyodorov \& Keating concerning the extreme value statistics, moments of
moments, connections to Gaussian Multiplicative Chaos, and explicit formulae
derived from the theory of symmetric functions.
