The article reviews recent theoretical and experimental advances in the study of many-body localized (MBL) systems, focusing on the new perspective provided by entanglement and non-equilibrium experimental probes such as quantum quenches. MBL systems remain perfect insulators at non-zero temperatures, failing to thermalize and thus not describable by statistical mechanics. The review highlights the emergence of robust integrability in MBL systems, characterized by an extensive set of quasi-local integrals of motion (LIOMs), which provide an alternative channel of evolution for the system. This integrability explains the failure of thermalization and predicts dynamical properties such as the spreading of quantum entanglement, the behavior of local observables, and the response to external dissipative processes. MBL systems can also exhibit eigenstate transitions and quantum orders forbidden in thermodynamic equilibrium, such as localization-protected quantum orders and infinite temperature breaking of discrete symmetries. The review discusses the quantum-to-classical transition between MBL and ergodic phases, as well as anomalous transport near this transition. Experimentally, synthetic quantum systems, well-isolated from external thermal reservoirs, provide platforms for realizing the MBL phase. Recent experiments with ultracold atoms, trapped ions, superconducting qubits, and quantum materials have observed signatures of many-body localization. The article concludes by outlining outstanding challenges and future research directions in the field.The article reviews recent theoretical and experimental advances in the study of many-body localized (MBL) systems, focusing on the new perspective provided by entanglement and non-equilibrium experimental probes such as quantum quenches. MBL systems remain perfect insulators at non-zero temperatures, failing to thermalize and thus not describable by statistical mechanics. The review highlights the emergence of robust integrability in MBL systems, characterized by an extensive set of quasi-local integrals of motion (LIOMs), which provide an alternative channel of evolution for the system. This integrability explains the failure of thermalization and predicts dynamical properties such as the spreading of quantum entanglement, the behavior of local observables, and the response to external dissipative processes. MBL systems can also exhibit eigenstate transitions and quantum orders forbidden in thermodynamic equilibrium, such as localization-protected quantum orders and infinite temperature breaking of discrete symmetries. The review discusses the quantum-to-classical transition between MBL and ergodic phases, as well as anomalous transport near this transition. Experimentally, synthetic quantum systems, well-isolated from external thermal reservoirs, provide platforms for realizing the MBL phase. Recent experiments with ultracold atoms, trapped ions, superconducting qubits, and quantum materials have observed signatures of many-body localization. The article concludes by outlining outstanding challenges and future research directions in the field.