We study the heat capacity and neutrino emission reactions (direct and modified Urca processes, nucleon-nucleon bremsstrahlung, Cooper pairing of nucleons) in the supranuclear density matter of neutron star cores with superfluid neutrons and protons. Various superfluidity types are analysed (singlet-state pairing and two types of triplet-state pairing, without and with gap nodes at the nucleon Fermi surface). The results are used for cooling simulations of isolated neutron stars. Both the standard cooling and the cooling enhanced by the direct Urca process are strongly affected by nucleon superfluidity. Comparison of the cooling theory of isolated neutron stars with observations of their thermal radiation may give stringent constraints on the critical temperatures of the neutron and proton superfluidities in the neutron star cores.