A survey of moment bounds for $ζ(s)$: from Heath Brown's work to the present

In this expository article, we review some of the ideas behind the work of Heath-Brown (D.~R. Heath-Brown, {\em Fractional moments of the Riemann Zeta Function}, J. Lond. Math. Soc., (2), {\bf 24}, no.1, (1981), 65--78) on upper and lower bounds for moments of the Riemann zeta-function, as well as the impact this work had on subsequent developments in the field. We survey recent results on the topic, which essentially recover the expected rate of growth for all moments - unconditionally for small moments and conditionally on the Riemann hypothesis for all larger moments.