The Markoff equation is x² + y² + z² = 3xyz. For just a single equation, it turns up remarkably often in a wide variety of mathematical disciplines. These include Diophantine approximation, hyperbolic geometry, algebraic geometry, and graph theory. We will discuss these appearances of the Markoff equation, with a focus on graph theory and the recent results surrounding the Baragar and Bourgain-Gamburd-Sarnak conjectures.