The detection of gravitational waves has opened a new era of scientific discovery, as it permits a new kind of observation of the cosmos, quite different from electromagnetic and particle observations. In this talk I will review the gravitational-wave signals detected up to now by LIGO and Virgo, and discuss the theoretical groundwork that allows to identify and interpret those signals. I will also highlight how those new astronomical messengers are unveiling the properties of the most extreme astrophysical objects in the universe and probe fundamental physics.