In the first part of these lectures I will present certain, by now classic, results on local and global well-posedness for the NLS via Strichartz estimates. In the second part I will show how one can use randomization of the initial data to prove well-posedness almost surely even when the problem lack enough regularity for a more deterministic approach. In this context I will also introduce certain Gibbs type measures and how their invariance can be used in certain cases to extend local solutions to a global ones.