Many-Body Localization in the Age of Classical Computing This review discusses the current understanding of many-body localization (MBL) and its implications for thermalization in disordered quantum systems. MBL is a phenomenon where interacting quantum systems, under strong disorder, fail to thermalize, even in the asymptotic limit of infinite system size and time. The eigenstate thermalization hypothesis (ETH) provides a framework for understanding thermalization in isolated quantum systems, but MBL challenges this by showing that thermalization does not occur in certain conditions. The review highlights recent numerical studies that have clarified the status of the MBL phase and the challenges in interpreting results. It discusses the difficulties in numerical approaches to MBL, including the limitations of single-parameter scaling and the need for more refined scaling procedures. The review also emphasizes the persistent finite size drifts towards ergodicity in disordered many-body systems, which are related to continuous inching towards ergodicity and non-vanishing transport even at strong disorder. The review explores various approaches to understanding the MBL phase, including the role of local integrals of motion (LIOMs) and the implications of the MBL phase transition. It discusses the relation of thermalization and transport in many-body systems, highlighting the importance of understanding the dynamics of disordered many-body systems. The review also considers various extensions of the MBL phenomenology, including other models of MBL, quasiperiodic potentials, and long-range interactions. The review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations. The review emphasizes the need for further research to understand the dynamics of disordered many-body systems and the implications of the MBL phase for the broader field of quantum many-body systems.Many-Body Localization in the Age of Classical Computing This review discusses the current understanding of many-body localization (MBL) and its implications for thermalization in disordered quantum systems. MBL is a phenomenon where interacting quantum systems, under strong disorder, fail to thermalize, even in the asymptotic limit of infinite system size and time. The eigenstate thermalization hypothesis (ETH) provides a framework for understanding thermalization in isolated quantum systems, but MBL challenges this by showing that thermalization does not occur in certain conditions. The review highlights recent numerical studies that have clarified the status of the MBL phase and the challenges in interpreting results. It discusses the difficulties in numerical approaches to MBL, including the limitations of single-parameter scaling and the need for more refined scaling procedures. The review also emphasizes the persistent finite size drifts towards ergodicity in disordered many-body systems, which are related to continuous inching towards ergodicity and non-vanishing transport even at strong disorder. The review explores various approaches to understanding the MBL phase, including the role of local integrals of motion (LIOMs) and the implications of the MBL phase transition. It discusses the relation of thermalization and transport in many-body systems, highlighting the importance of understanding the dynamics of disordered many-body systems. The review also considers various extensions of the MBL phenomenology, including other models of MBL, quasiperiodic potentials, and long-range interactions. The review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations. The review emphasizes the need for further research to understand the dynamics of disordered many-body systems and the implications of the MBL phase for the broader field of quantum many-body systems.