In these three lectures I will give an overview of the DiPerna-Lions theory for transport equations and ODEs with Sobolev coefficients, including the developments of the last two decades, namely Ambrosio's extension to the case of BV coefficients, Lagrangian estimates and Bressan's mixing conjecture, near incompressibility of the flow versus bounds on the divergence of the field, the role of summability and the impact of convex integration techniques. I will try to cover the general ideas of several results, without going into the technical details, and I will give a special emphasis on open problems.