Better than squareroot cancellation in number theory

We give a short survey of the phenomenon of better than squareroot cancellation, specifically as it applies to averages of multiplicative character sums (such as $\frac{1}{r-1}{\sum }_{\chi \text{mod}r}\mathrm{\mid }{\sum }_{n\le x}\chi (n){\mathrm{\mid }}^{2q}$) thanks to their connection with so-called multiplicative chaos. We focus on the number theoretic aspects of the arguments, and also touch on some possible applications.