On large scales, the peculiar velocities of galaxies encode rich cosmological information, providing a complementary probe of the growth of structure and gravity. Along the line of sight, these velocities can be inferred through Doppler shifts. Two major observational effects allow us to extract this information: redshift-space distortions in galaxy clustering and the imprint of the bulk motions of ionised gas on the cosmic microwave background through the kinematic Sunyaev–Zel’dovich effect. In the transverse direction, measuring transverse motions is more challenging, but new approaches have emerged that allow us to probe them indirectly. I will summarise some recent analyses in this research area and demonstrate how we can do better.