Pairing gaps in neutron matter need to be computed in a wide range of densities to address open questions in neutron-star phenomenology. Traditionally, the Bardeen-Cooper-Schrieffer approach has been used to compute gaps from bare nucleon-nucleon interactions. Here we incorporate the influence of short-and long-range correlations in the pairing gaps. Short-range correlations are treated, including the appropriate fragmentation of single-particle states, and substantially suppress the gaps. Long-range correlations dress the pairing interaction via density and spin modes and provide a relatively small correction. We use different interactions, some with three-body forces, as a starting point to control for any systematic effects. Results are relevant for neutron-star cooling scenarios, in particular in view of the recent observational data on Cassiopeia A.