Inspired by the General Relativity for many decades, experimental physicists and astronomers have a solid dream to detect gravitational waves (GWs) from mergers of black holes, which came true until the excellent performance of the Laser Interferometer Gravitational-Wave Observatory (LIGO) at hundreds Hz. Nano-Hz GWs are expected to be radiated by close-binaries of supermassive black holes (CB-SMBHs;defined as those with separations less than ～0.1 pc) formed during galaxy mergers and detected through the Pulsar Timing Array (PTA) technique. As of the writing, there remains no nano-Hz GWs detection.Searching for CB-SMBHs is also observationally elusive though there exist a number of possible candidates.In this review, we focus on observational signatures of CB-SMBHs from theoretic expectations, simulations and observations. These signatures appear in energy distributions of multiwavelength continuum, long term variations of continuum, jet morphology, reverberation delay maps and spectroastrometry of broad emission lines, AGN type transitions between type-1 and type-2 (changing-look), and gaseous dynamics of circumbinary disks, etc. Unlike hundred-Hz GWs from stellar mass black hole binaries, the waveform chirping of nano-Hz GWs is too slow to detect in a reasonable human timescale. We have to resort to electromagnetic observations to measure orbital parameters of CB-SMBHs to test nano-Hz GW properties.Reverberation mapping is a powerful tool for probing kinematics and geometry of ionized gas in the gravitational well of SMBHs (single or binary) and therefore provides a potential way to determine orbital parameters of CB-SMBHs. In particular, a combination of reverberation mapping with spectroastrometry(realized at the Very Large Telescope Interferometer) will further reinforce this capability. The Atacama Large Millimeter/submillimeter Array (ALMA) and the forthcoming Square Kilometre Array (SKA) are suggested to reveal dynamics of circumbinary disks through molecular emission lines.