  We discuss two-sided bounds for moments and tails of quadratic forms in
Gaussian random variables with values in Banach spaces. We state a natural
conjecture and show that it holds up to additional logarithmic factors.
Moreover in a certain class of Banach spaces (including $L_r$-spaces) these
logarithmic factors may be eliminated. As a corollary we derive upper bounds
for tails and moments of quadratic forms in subgaussian random variables, which
extend the Hanson-Wright inequality.
