Explore anomalous diffusive fluctuations in low-dimensional quantum systems in this 43-minute lecture by Romain Vasseur, part of the "Stability of Quantum Matter in and Out of Equilibrium at Various Scales" program at the International Centre for Theoretical Sciences. Delve into the complex world of many-body quantum systems, their coherent time evolution, and stability under external drives. Examine mechanisms constraining many-body dynamics, including Many-body localization, Floquet MBL, Dynamical Freezing, and Hilbert Space Fragmentation. Gain insights into novel phases of quantum matter, drawing from diverse disciplines such as Condensed Matter Physics, Statistical Mechanics, and Quantum Field Theory. Understand the implications for quantum device stability and the broader field of quantum technologies.