Landau's idea of classifying phases of matter in terms of symmetry breaking is a cornerstone of modern physics. In his pioneering work Wilczek  argued that an autonomous system can break time translation symmetry, thus realizing what he named as time crystals. Their existence has been confirmed very recently in a number of experiments. I will review the field focusing in particular on the so called Floquet time-crystals arising in many-body systems that are periodically driven.

Prof. Rosario Fazio obtained his PhD degree from the University of Catania in 1989, where he became Assistant Professor in 1990 and Associate Professor in 1998. He moved in 2001 to the Scuola Normale Superiore in Pisa, and in 2005 became full Professor in SISSA, Trieste. He returned to the Scuola Normale Superiore in 2008, of which he is still Professor of Condensed Matter Physics. In 2016 he became the Head of the Condensed Matter and Statistical Physics Sector of ICTP. Currently, he is also a Visiting Professor at the Center for Quantum Technologies, National University of Singapore. His main research interests are on quantum transport in nanostructures, superconductivity (Josephson junction arrays, superconducting nanocircuits), quantum simulators (cold atoms in optical lattices, QED-arrays), non-equilibrium dynamics of many-body systems, and quantum information processing.