In this contribution we discuss how neutron stars are produced and retained in globular clusters, outlining the most important dynamical channels and evolutionary events that affect the population of mass-transferring binaries with neutron stars and result in the formation of recycled pulsars. We confirm the importance of electron-capture supernovae in globular clusters as the major supplier of retained neutron stars. By comparing the observed millisecond pulsar population and the results obtained from simulations, we discuss several constraints on the evolution of mass-transferring systems. In particular, we find that in our cluster model the following mass-gaining events create populations of MSPs that do not match the observations (with respect to binary periods and companion masses or the number of produced systems) and therefore likely do not lead to NSs spun up to millisecond periods: (i) accretion during a common envelope event with a NS formed through accretion-induced collapse, and (ii) mass transfer from a WD donor. By restricting ourselves to the evolutionary and dynamical paths that most likely lead to neutron star recycling, we obtain good agreement between our models and the numbers and characteristics of observed millisecond pulsars in the clusters Terzan 5 and 47 Tuc.