We review the long standing problem of superfluid pairing in pure neutron matter. For the s-wave pairing, we summarize the state of the art of many-body approaches including different nn interactions, medium polarization, short-range correlations and BCS–BEC crossover effects, and compare them with quantum Monte Carlo results at low densities. We also address pairing in the p-wave, which appears at higher densities and, hence, has large uncertainties due to the poorly constrained interactions, medium effects and many-body forces. © 2021, The Author(s), under exclusive licence to EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.