Fast radio bursts (FRBs) are brief, highly dispersed bursts detected in the radio band that originate from cosmological distances. The only such event detected in the Milky Way Galaxy, FRB 20200428D—which was associated with an X-ray burst emitted by a magnetar named SGR J1935+2154—revealed the first case of a multiwavelength counterpart of an FRB. Counterparts in other wavelengths accompanying or following FRBs, as well as the bright emission associated with the progenitor of the FRB engine, have been proposed in various FRB models, but no robust detection has been made so far. In general, FRBs as we know them are not favored multimessenger emitters. Nonetheless, possible neutrino and gravitational wave emission signals associated with FRBs or FRB-like events have been discussed in the literature. Here I review these suggested multiwavelength and multimessenger counterparts of FRBs or FRB-like events and the observational progress in searching for these signals. Topics include multiwavelength (X-rays, γ-rays, optical) emission and neutrino emission from FRBs within the framework of the magnetar source models and possible FRB-like events associated with gravitational waves.