On 2015 September 14, the first observation of gravitational-waves by the Advanced Laser Interferometer Gravitational-wave Observatory detectors concluded a long scientific quest, which began 100 years before with Einstein’s prediction of their existence. This detection opened a new exploration of the Universe making it possible to access the properties of space-time at extreme regime, to probe the properties of compact objects (binary systems of neutron stars and stellar-mass black holes), and investigate their formation and evolution. On August 17, 2017, the first observation of gravitational waves from the inspiral and merger of a binary neutron-star system by the Advanced LIGO and Virgo network, followed 1.7 s later by a weak short gamma-ray burst detected by the Fermi and INTEGRAL satellites initiated the most extensive world-wide observing campaign which led to the detection of multi-wavelength electromagnetic counterparts. Multi-messenger discoveries are unveiling the rich physics of most energetic transient phenomena in the sky, probing relativistic astrophysics, nuclear physics, nucleosynthesis, and cosmology. Here, we give an overview of the recent gravitational-wave and multi-messenger discoveries, and the perspectives for the future.