Sums of Dirichlet characters ∑n≤xχ(n)∑n≤xχ(n) (where χχ is a character modulo some prime rr, say) are one of the best studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments 1r−1∑χ mod r|∑n≤xχ(n)|2q1r−1∑χ mod r|∑n≤xχ(n)|2q, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when 0≤q≤10≤q≤1. I will focus mainly on the number theoretic issues arising.