The method of Darboux is a classical integration method for finding exact solutions to scalar, second-order partial differential equations (PDE) solely through the integration of ordinary differential equations – PDE which can be solved in this way are called Darboux integrable.
Here, we provide historical context to the development of this method, illustrate the classical approach with examples, and introduce a modern approach to Darboux integrability using differential-geometric and Lie-theoretic machinery. We then highlight several examples and applications from the modern approach.