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Most of the stars in our Galaxy including our Sun, move in a disk-like component and give the Milky Way its characteristic appearance on the night sky. As in all fields in science, motions can be used to reveal the underlying forces, and in the case of disk stars they provide important diagnostics on the structure and history of the Galaxy. But because of the challenges involved in measuring stellar motions, samples have so far remained limited in their number of stars, precision and spatial extent. This has changed dramatically with the second Data Release of the Gaia mission which has just become available. Here we report that the phase space distribution of stars in the disk of the Milky Way is full of substructure with a variety of morphologies, most of which have never been observed before. This includes shapes such as arches and shells in velocity space, and snail shells and ridges when spatial and velocity coordinates are combined. The nature of these substructures implies that the disk is phase mixing from an out of equilibrium state, and that the Galactic bar and/or spiral structure are strongly affecting the orbits of disk stars. Our analysis of the features leads us to infer that the disk was perturbed between 300 and 900 Myr ago, which matches current estimations of the previous pericentric passage of the Sagittarius dwarf galaxy. The Gaia data challenge the most basic premise of stellar dynamics of dynamical equilibrium, and show that modelling the Galactic disk as a time-independent axisymmetric component is definitively incorrect. These findings mark the start of a new era when, by modelling the richness of phase space substructures, we can determine the gravitational potential of the Galaxy, its time evolution and the characteristics of the perturbers that have most influenced our home in the Universe.