In 2034 the Laser Interferometer Space Antenna (LISA) will detect the coalescence of massive black hole binaries (MBHBs) from 10^5 to 10^7 solar mass up to z~10. The gravitational wave (GWs) signal is expected to be accompanied by a powerful electromagnetic (EM) counterpart, from radio to X-ray, generated by the gas accreting on the binary. If LISA locates the MBHB merger within an error box <10 square degree, EM telescopes can be pointed in the same portion of the sky to detect the emission from the last stages of the MBHB orbits or the very onset of the nuclear activity, paving the way to test the nature of gas in a rapidly changing space-time. Moreover, an EM counterpart will allow independent measurements of the source redshift which, combined with the luminosity distance estimate from the GW signal, will lead to exquisite tests on the expansion of the Universe as well as on the velocity propagation of GWs. In this talk, I present some recent results on the expected rates of MBHBs counterparts detectable jointly by LISA and EM facilities and the possible applications in cosmology.

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