Title: Non-equilibrium and information: the role of cross-correlations Abstract: I will discuss the relevance of information contained in cross-correlations among different degrees of freedom, which is crucial in non-equilibrium systems. In particular I consider a stochastic system where two degrees of freedom, in contact with two different thermostats, are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation relations are both related to the cross-correlation between the two variables. Such an information may be lost when single-variable reduced models are considered. In particular, in the case of linear couplings, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting the observation of the fluctuations response relations. I will conclude by describing an application to a many body system, namely a massive tracer in a driven granular gas.