<p>Plasma-based acceleration of electrons has produced high-energy beams at high accelerating gradients with a narrow energy spread and high efficiency both in experiments and simulations. It is now being considered as a complementary approach to the use of radiofrequency cavities in next-generation lepton accelerators. However, compared with electrons, plasma-based positron acceleration is at the present time much less advanced. Although high-gradient positron acceleration in a plasma has been achieved, we are one to three orders of magnitude away from delivering the high-quality positron beams needed for a future high-energy linear collider. Here we review the status of plasma-based acceleration of electrons and positrons and discuss the prospects for substantial progress towards developing the positron arm of a plasma-based electron–positron linear collider in the next decade.</p>