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We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x}f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicativefunction, and $0 \leq q \leq 1$. In the Steinhaus case, this is equivalent todetermining the order of $\lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T}|\sum_{n \leq x} n^{-it}|^{2q} dt$.  In particular, we find that $\mathbb{E}|\sum_{n \leq x} f(n)| \asymp\sqrt{x}/(\log\log x)^{1/4}$. This proves a conjecture of Helson that oneshould have better than squareroot cancellation in the first moment, anddisproves counter-conjectures of various other authors. We deduce someconsequences for the distribution and large deviations of $\sum_{n \leq x}f(n)$.  The proofs develop a connection between $\mathbb{E}|\sum_{n \leq x}f(n)|^{2q}$ and the $q$-th moment of a critical, approximately Gaussian,multiplicative chaos, and then establish the required estimates for that. Weinclude some general introductory discussion about critical multiplicativechaos to help readers unfamiliar with that area.

Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos