  The classical double copy relates exact solutions of gauge, gravity and other
theories. Although widely studied, its origins and domain of applicability have
remained mysterious. In this letter, we show that a particular incarnation -
the Weyl double copy - can be derived using well-established ideas from twistor
theory. As well as explaining where the Weyl double copy comes from, the
twistor formalism also shows that it is more general than previously thought.
