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The detection of gravitational waves by the ground based interferometers LIGO and Virgo has provided an entirely new instrument to observe and investigate the universe we live in, challenging or confirming our beliefs on several crucial issues: how does gravity behave in the strong field regime, as near a black hole horizon; how does matter behave at the supranuclear densities which are typical of the inner core of neutron stars; how do gravitating structures evolve to form compact objects which eventually produce binary coalescence as those which have been, or will be, observed. The advent of LISA will further enhance our capability to investigate fundamental physics with gravitational waves.
These issues will be illustrated and discussed.