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We prove conjecturally sharp upper bounds for the Dirichlet character moments$\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$,where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real.In particular, if both $x$ and $r/x$ tend to infinity with $r$ then$\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)| =o(\sqrt{x})$, and so the sums $\sum_{n \leq x} \chi(n)$ typically exhibit"better than squareroot cancellation". We prove analogous better thansquareroot bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x}n^{it}|^{2q} dt$ of zeta sums; of Dirichlet theta functions $\theta(1,\chi)$;and of the sums $\sum_{n \leq x} h(n) \chi(n)$, where $h(n)$ is any suitablybounded multiplicative function (for example the M\"{o}bius function $\mu(n)$).  The proofs depend on similar better than squareroot cancellation phenomenafor low moments of random multiplicative functions. An important ingredient isa reorganisation of the conditioning arguments from the random case, so thatone only needs to "condition" on a small collection of fairly short primenumber sums. The conditioned quantities arising can then be well approximatedby twisted second moments, whose behaviour is the same for character and zetasums as in the random case.

The typical size of character and zeta sums is $o(\sqrt{x})$