Many-body localization (MBL) is a phenomenon where quantum systems fail to thermalize due to strong disorder, leading to a robust insulating phase at non-zero temperatures. This phase exhibits a new form of integrability, characterized by an extensive set of quasi-local integrals of motion (LIOMs), which explains the breakdown of thermalization. MBL systems show unique properties, such as area-law entanglement scaling and the emergence of quantum orders not seen in thermal equilibrium. Recent theoretical and experimental studies have revealed that MBL systems can exhibit eigenstate transitions and anomalous transport near the delocalization transition. Experiments with ultracold atoms, trapped ions, superconducting circuits, and quantum materials have observed signatures of MBL. The MBL phase is distinguished by its robustness against thermalization and its ability to host non-thermal quantum states. Theoretical advances include the development of new numerical and analytical approaches to study MBL, while experimental developments highlight the potential of synthetic quantum systems for realizing MBL. The MBL phase represents a new class of quantum states that challenge conventional statistical mechanics and offer insights into non-equilibrium dynamics and quantum many-body systems.Many-body localization (MBL) is a phenomenon where quantum systems fail to thermalize due to strong disorder, leading to a robust insulating phase at non-zero temperatures. This phase exhibits a new form of integrability, characterized by an extensive set of quasi-local integrals of motion (LIOMs), which explains the breakdown of thermalization. MBL systems show unique properties, such as area-law entanglement scaling and the emergence of quantum orders not seen in thermal equilibrium. Recent theoretical and experimental studies have revealed that MBL systems can exhibit eigenstate transitions and anomalous transport near the delocalization transition. Experiments with ultracold atoms, trapped ions, superconducting circuits, and quantum materials have observed signatures of MBL. The MBL phase is distinguished by its robustness against thermalization and its ability to host non-thermal quantum states. Theoretical advances include the development of new numerical and analytical approaches to study MBL, while experimental developments highlight the potential of synthetic quantum systems for realizing MBL. The MBL phase represents a new class of quantum states that challenge conventional statistical mechanics and offer insights into non-equilibrium dynamics and quantum many-body systems.