These lectures concern the Euler and Navier-Stokes equations, which are models for incompressible fluid flow. The focus is the mathematical analysis of these partial differential equations. We will discuss the main steps and ideas for local-in-time well-posedness of classical solutions, the problem of singularity formation and the Beale-Kato-Majda criterion and, finally, the issue of vortex stretching in three dimensions. We then begin discussing weak solutions. We will explain the construction and proof of global-in-time existence of Leray-Hopf weak solutions of the 3D Navier-Stokes equations and the weak-strong uniqueness theorem due to Prodi-Serrin. Lastly we consider the special case of 2D flows and we will discuss results for weak solutions, in particular the Yudovich uniqueness theorem. In view of time constraints it will only be possible to give rough sketches of proofs but we will provide references where further details can be found. These lectures aim to serve as background material for the remaining lectures of the Summer School.