Explore the intricacies of non-stationary quantum many-body dynamics in this first lecture by Berislav Buca, part of the "Periodically and Quasi-periodically Driven Complex Systems" program at the International Centre for Theoretical Sciences. Delve into advanced topics in quantum physics, including time-dependent Hamiltonians, Many Body Localization, open quantum systems, and gauge theory of coupling between matter and radiation. Gain insights into cutting-edge research areas such as terahertz and infrared driven quantum matter, shaken optical lattices and cold atoms, Floquet time crystals, and localization-delocalization physics in quasi-periodically driven systems. This 59-minute lecture offers a valuable opportunity to engage with complex concepts at the intersection of theory and experiment in quantum physics.