Twistor theory is a non-local correspondence between light-cones in space-time and points in certain complex manifold. It was proposed by Roger Penrose as a way towards quantum gravity. Quite unexpectedly, the theory led to a novel way of solving differential equations in mathematical physics and geometry. I shall present a self-contained and elementary overview of the subject. I shall also explain a twistor solution to the Liouville problem which remained open for more than 120 years.