Explore quantum many-body dynamics, thermalization, and integrability-breaking in this comprehensive lecture by David Huse, part of the "Stability of Quantum Matter in and out of Equilibrium at Various Scales" program. Delve into the behavior of coherent quantum systems, examining their stability under external drives and various mechanisms constraining many-body dynamics in Hilbert space. Investigate cutting-edge concepts like Many-body localization (MBL), Floquet MBL, Dynamical Freezing, and Hilbert Space Fragmentation, drawing from diverse fields such as Condensed Matter Physics, Statistical Mechanics, and Quantum Field Theory. Gain insights into the state-of-the-art tools and concepts relevant to understanding many-body stability problems, with potential applications in quantum device operation and control of quantum systems.