The origin and phenomenology of the Fast Radio Burst (FRB) remains unknown despite more than a decade of efforts. Though several models have been proposed to explain the observed data, none is able to explain alone the variety of events so far recorded. The leading models consider magnetars as potential FRB sources. The recent detection of FRBs from the galactic magnetar SGR J1935+2154 seems to support them. Still, emission duration and energetic budget challenge all these models. Like for other classes of objects initially detected in a single band, it appeared clear that any solution to the FRB enigma could only come from a coordinated observational and theoretical effort in an as wide as possible energy band. In particular, the detection and localisation of optical/NIR or/and high-energy counterparts seemed an unavoidable starting point that could shed light on the FRB physics. Multiwavelength (MWL) search campaigns were conducted for several FRBs, in particular for repeaters. Here we summarize the observational and theoretical results and the perspectives in view of the several new sources accurately localised that will likely be identified by various radio facilities worldwide. We conclude that more dedicated MWL campaigns sensitive to the millisecond--minute timescale transients are needed to address the various aspects involved in the identification of FRB counterparts. Dedicated instrumentation could be one of the key points in this respect. In the optical/NIR band, fast photometry looks to be the only viable strategy. Additionally, small/medium size radiotelescopes co-pointing higher energies telescopes look a very interesting and cheap complementary observational strategy.
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