Rare kaon decays are among the most sensitive probes of both heavy and light New Physics (NP) beyond the Standard Model (SM) thanks to the high precision of the SM predictions, to the availability of very large datasets and to the relatively simple decay topologies. The NA62 experiment at CERN SPS is a multi-purpose high-intensity kaon decay experiment, that carries out a broad rare-decay and hidden-sector physics program and has collected a large sample of $K^+$ decays during Run1 (2016-2017-2018) and the ongoing Run2 (started in 2021). Recent NA62 results on searches in Run1 data for hidden-sector mediators in kaon decays are here presented. NA62 can also be run as a beam-dump experiment by removing the kaon production target and moving the upstream collimators into a "closed" position. In this configuration, 400 GeV/c protons from the SPS are dumped on an absorber. NP states feebly interacting with SM particles may be produced in the dump and reach the fiducial decay volume of the NA62 detector, that begins 80 m downstream of the absorber. More than $10^{17}$ protons on target have been collected in beam-dump mode in 2021. The results from the analysis of beam-dump data are also reported, with particular emphasis on dark photon and axion-like particle models.