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

The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwi\l\l, and Soundararajan]