We discuss and review some main results on the derivation of Euler and Navier-Stokes equations from microscopic dynamics: we restrict our attention to two kinds of microscopic dynamics: the Boltzmann equation and some stochastic particle systems. We introduce the concept of hydrodynamic limit and show how to prove in such a limit the convergence of kinetic systems to macroscopic equations in various situations such as in presence of an external force or in a bounded domain both in time-dependent or stationary case. We introduce systems of particles moving according to some natural stochastic dynamics, in the real space or on the lattice, and discuss the relevance of a new probabilistic method, called non-gradient method, for proving the hydrodynamic limit on the diffusive scale and for getting the transport coefficients in terms of the Green-Kubo formulas.