  We determine to leading order the maximum of the characteristic polynomial
for Wigner matrices and $\beta$-ensembles. In the special case of
Gaussian-divisible Wigner matrices, our method provides universality of the
maximum up to tightness. These are the first universal results on the
Fyodorov--Hiary--Keating conjectures for these models, and in particular answer
the question of optimal rigidity for the spectrum of Wigner matrices.
  Our proofs combine dynamical techniques for universality of eigenvalue
statistics with ideas surrounding the maxima of log-correlated fields and
Gaussian multiplicative chaos.
