The aim of this talk is to review some basic questions arising from nonequilibrium statistical mechanics and to explain why large deviations offer a suitable mathematical framework to rephrase these questions. In the simplified setting of stochastic models, we will compute the large deviations of the heat current flowing through a diffusive system maintained off equilibrium by two heat baths at unequal temperatures. We will also discuss the dynamical phase transitions which may occur for some models and the structure of the long range correlations in systems maintained off equilibrium