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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 orRademacher 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. Inthe 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 consequencesfor the large deviations of $\sum_{n \leq x} f(n)$.  The proofs use various tools, including hypercontractive inequalities, toconnect $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of anEuler product integral. When $q$ is large, it is then fairly easy to analysethis integral. When $q$ is close to 1 the analysis seems to require subtlerarguments, including Doob's $L^p$ maximal inequality for martingales.

Moments of random multiplicative functions, II: High moments