  This is the text to accompany my Bourbaki seminar from 30th March 2019, on
the maximum size of the Riemann zeta function in "almost all" intervals of
length 1 on the critical line. It surveys the conjecture of
Fyodorov--Hiary--Keating on the behaviour of this typical maximum, as well as
recent progress towards the conjecture by Najnudel and by
Arguin--Belius--Bourgade--Radziwi\l\l--Soundararajan. There is also some
general background discussion of the value distribution and large values of
zeta.
