A partial differential equation looks complicated – it mixes changes in time and space all at once. What if we could turn it into a big system of ordinary differential equations, solve those with powerful, off‑the‑shelf ODE tools, and then stitch the pieces back together? That is exactly what the method of lines (MOL) does. In this chapter we will see how MOL works, meet some clever time integrators (like the matrix exponential), and even use MOL to price financial options with early‑exercise features and moving boundaries.