Search for dark sector physics with NA64

The NA64 experiment consists of two detectors which are planned to be located at the electron (NA64e) and muon (NA64$\mu$) beams of the CERN SPS and start operation after the LHC long-stop 2 in 2021. Its main goals include searches for dark sector physics - particularly light dark matter (LDM), visible and invisible decays of dark photons ($A'$), and new light particles that could explain the $^8$Be and $g_{\mu}-2$ anomalies. Here we review these physics goals, the current status of NA64 including recent results and perspectives of further searches, as well as other ongoing or planned experiments in this field. The main theoretical results on LDM, the problem of the origin of the $\gamma-A'$ mixing term and its connection to loop corrections, possible existence of a new light $Z'$ coupled to $L_\mu-L_\tau$ current are also discussed.