  This is a write-up of some lectures I gave in the Fall of 2021 at the Fields
Institute in Toronto, as part of the Thematic Programme on Trends in Pure and
Applied Model Theory. The goal of the module was to give a quick introduction
to the model theory of differential fields that puts differential-algebraic
geometry at the center. I focus here on the birational geometry of algebraic
vector fields and more generally $D$-varieties in the sense of Buium.
