  We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x}
f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or
Rademacher random multiplicative function, for all real $1 \leq q \leq
\frac{c\log x}{\log\log x}$.
  In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} =
e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In
the Rademacher case, we find a transition in the behaviour of the moments when
$q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by
"orthogonal" rather than "unitary" behaviour. We also deduce some consequences
for the large deviations of $\sum_{n \leq x} f(n)$.
  The proofs use various tools, including hypercontractive inequalities, to
connect $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of an
Euler product integral. When $q$ is large, it is then fairly easy to analyse
this integral. When $q$ is close to 1 the analysis seems to require subtler
arguments, including Doob's $L^p$ maximal inequality for martingales.
