Through the random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the freezing scenario. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions. I will explain these developments, in particular joint work with Arguin and Radziwill.