Explore the fascinating world of Floquet many-body physics in this comprehensive lecture by Roderich Moessner. Delve into the complex interplay between the eigenstate thermalization hypothesis (ETH) and spatiotemporal order in periodically driven quantum systems. Gain insights into the stability of quantum matter under external drives, and discover emerging concepts like Floquet many-body localization, dynamical freezing, and Hilbert space fragmentation. Examine the interdisciplinary nature of this field, drawing from condensed matter physics, statistical mechanics, and quantum field theory. Understand the potential applications in quantum technologies and the control of quantum systems. Suitable for senior PhD students, postdocs, and faculty members working in related fields, this lecture is part of a broader program on the stability of quantum matter in and out of equilibrium at various scales.