I will discuss results by R. Latala concerning tail behaviour of multivariate polynomials in independent Gaussian variables and show how when combined with classical functional inequalities they give estimates for polynomials and more generally smooth functions with bounded derivatives of higher order for a more general class of non-necessarily product measures. I will also present similar inequalities for polynomials of general sequences of independent subgaussian random variables, which do not possess concentration properties for Lipschitz functions. Based on joint work with P. Wolff (University of Warsaw).