Rare kaon decays are among the most sensitive probes of both heavy and light physics beyond the Standard Model (SM) thanks to the high precision of the SM predictions, the availability of very large datasets and the relatively simple decay topologies. NA62 is the CERN multi-purpose fixed-target kaon decay experiment, carring out a broad rare decay and hidden sector physics program. Recent NA62 searches for lepton flavor and lepton number violation and for the production of hidden sector mediators in $K^+$ decays, along with their future prospects, are presented.