We show that spiral waves in galaxy discs churn the stars and gas in a manner that largely preserves the overall angular momentum distribution and leads to little increase in random motion. Changes in the angular momenta of individual stars are typically as large as ∼50 per cent over the lifetime of the disc. The changes are concentrated around the corotation radius for an individual spiral wave, but since transient waves with a wide range of pattern speeds develop in rapid succession, the entire disc is affected. This behaviour has profound consequences for the metallicity gradients with radius in both stars and gas, since the interstellar medium is also stirred by the same mechanism. We find observational support for stirring, propose a simple model for the distribution of stars over metallicity and age, and discuss other possible consequences.