  The quest for binary and dual supermassive black holes (SMBHs) at the dawn of
the multi-messenger era is compelling. Detecting dual active galactic nuclei
(AGN) -- active SMBHs at projected separations larger than several parsecs --
and binary AGN -- probing the scale where SMBHs are bound in a Keplerian binary
-- is an observational challenge. The study of AGN pairs (either dual or
binary) also represents an overarching theoretical problem in cosmology and
astrophysics. The AGN triggering calls for detailed knowledge of the
hydrodynamical conditions of gas in the imminent surroundings of the SMBHs and,
at the same time, their duality calls for detailed knowledge on how galaxies
assemble through major and minor mergers and grow fed by matter along the
filaments of the cosmic web. This review describes the techniques used across
the electromagnetic spectrum to detect dual and binary AGN candidates and
proposes new avenues for their search. The current observational status is
compared with the state-of-the-art numerical simulations and models for
formation of dual and binary AGN. Binary SMBHs are among the loudest sources of
gravitational waves (GWs) in the Universe. The search for a background of GWs
at nHz frequencies from inspiralling SMBHs at low redshifts, and the direct
detection of signals from their coalescence by the Laser Interferometer Space
Antenna in the next decade, make this a theme of major interest for
multi-messenger astrophysics. This review discusses the future facilities and
observational strategies that are likely to significantly advance this
fascinating field.
