The center theme of this talk is the effect that randomization on the initial data set has on questions of global well-posedness for a variety of evolution equations. I will start by recalling the notion of Gibbs measure for certain periodic dispersive equations in Hamiltonian form, a work that goes back to Lebowitz-Rose-Speer. I will continue with a short summary of the work of Bourgain, who proved invariance of the Gibbs measure for certain NLS equation and an almost sure global well-posedness as a consequence. I will then continue by illustrating how randomization can be effectively used even when an Hamiltonian structure is not present and as a consequence a Gibbs measure cannot be defined. I will illustrate in this context results proved for example for the Navier-Stokes and wave equations in the supercritical regime.