We use a suite of numerical simulations to investigate the mechanisms and effects of radial migration of stars in disc galaxies such as the Milky Way (MW). An isolated, collisionless stellar disc with an MW-like scaleheight shows only the radial ‘blurring’ expected of epicyclic orbits. Reducing the disc thickness or adding gas to the disc substantially increases the level of radial migration, induced by interaction with transient spiral arms and/or a central bar. We also examine collisionless discs subjected to gravitational perturbations from a cosmologically motivated satellite accretion history. In the perturbed disc that best reproduces the observed properties of MW, 20 per cent of stars that end up in the solar annulus 7 kpc < R < 9 kpc started at R < 6 kpc, and 7 per cent started at R > 10 kpc. This level of migration would add considerable dispersion to the age–metallicity relation of solar neighbourhood stars. In the isolated disc models, the probability of migration traces the disc’s radial mass profile, but in perturbed discs migration occurs preferentially at large radii, where the disc is more weakly bound. The orbital dynamics of migrating particles are also different in isolated and perturbed discs: satellite perturbations drive particles to lower angular momentum for a given change in radius. Thus, satellite perturbations appear to be a distinct mechanism for inducing radial migration, which can operate in concert with migration induced by bars and spiral structures. We investigate correlations between changes in radius and changes in orbital circularity or vertical energy, identifying signatures that might be used to test models and distinguish radial migration mechanisms in chemodynamical surveys of the MW disc.