Remarkable experimental advances enabled creation of highly tunable and controllable quantum systems of ultracold atoms, trapped ions, and superconducting quantum bits. These platforms proved to be uniquely suited for probing non-equilibrium properties of interacting quantum systems. Based on the intuition from statistical mechanics, one may expect that an interacting system of many particles prepared in a non-equilibrium state will thermalize, settling to a state of thermodynamic equilibrium. Surprisingly, there are systems which do not follow this expectation. I will describe physical mechanisms of parametrically slow thermalization, and its complete breakdown. While thermalization is associated with quantum information scrambling, its absence can protect local quantum coherence, enabling non-equilibrium phenomena not envisioned within the framework of statistical mechanics. I will highlight recent theoretical insights into the remarkable physical properties of non-thermalising systems, based on the underlying patterns of quantum entanglement. I will finally describe recent attempts to identify new kinds of non-thermalizing dynamics, and propose a possible route towards developing a classification of dynamical universality classes in many-body systems, based on the notion of temporal entanglement.