Maxima of log-correlated fields: some recent developments*

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