  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 multiplicative
function, and $0 \leq q \leq 1$. In the Steinhaus case, this is equivalent to
determining 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 one
should have better than squareroot cancellation in the first moment, and
disproves counter-conjectures of various other authors. We deduce some
consequences 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. We
include some general introductory discussion about critical multiplicative
chaos to help readers unfamiliar with that area.
