  This paper is devoted to uniform versions of the Hanson-Wright inequality for
a random vector $X \in \mathbb{R}^n$ with independent subgaussian components.
The core technique of the paper is based on the entropy method combined with
truncations of both gradients of functions of interest and of the coordinates
of $X$ itself. Our results recover, in particular, the classic uniform bound of
Talagrand (1996) for Rademacher chaoses and the more recent uniform result of
Adamczak (2015), which holds under certain rather strong assumptions on the
distribution of $X$. We provide several applications of our techniques: we
establish a version of the standard Hanson-Wright inequality, which is tighter
in some regimes. Extending our techniques we show a version of the
dimension-free matrix Bernstein inequality that holds for random matrices with
a subexponential spectral norm. We apply the derived inequality to the problem
of covariance estimation with missing observations and prove an almost optimal
high probability version of the recent result of Lounici (2014). Finally, we
show a uniform Hanson-Wright type inequality in the Ising model under
Dobrushin's condition. A closely related question was posed by Marton (2003).
