Many-body localization (MBL) and thermalization in quantum statistical mechanics are reviewed, focusing on systems that fail to thermalize due to localization. The paper discusses the Eigenstate Thermalization Hypothesis (ETH), which posits that thermalization occurs when eigenstates of a system's Hamiltonian resemble thermal states. However, many-body localized systems, which do not thermalize, have eigenstates that violate the ETH, allowing them to retain information about initial conditions. These systems are of interest for quantum memory applications. The review highlights the distinction between thermalization and localization, emphasizing that while equilibrium statistical mechanics assumes thermalization, MBL systems exhibit unique phases and transitions. The paper also discusses the implications of MBL for quantum statistical mechanics, including the possibility of ordered phases and phase transitions at high energy and low spatial dimensions. It addresses open questions in the field, such as the nature of the MBL phase transition and the role of interactions in localization. The review concludes that MBL represents a new frontier in quantum statistical mechanics, with potential applications in quantum technologies.Many-body localization (MBL) and thermalization in quantum statistical mechanics are reviewed, focusing on systems that fail to thermalize due to localization. The paper discusses the Eigenstate Thermalization Hypothesis (ETH), which posits that thermalization occurs when eigenstates of a system's Hamiltonian resemble thermal states. However, many-body localized systems, which do not thermalize, have eigenstates that violate the ETH, allowing them to retain information about initial conditions. These systems are of interest for quantum memory applications. The review highlights the distinction between thermalization and localization, emphasizing that while equilibrium statistical mechanics assumes thermalization, MBL systems exhibit unique phases and transitions. The paper also discusses the implications of MBL for quantum statistical mechanics, including the possibility of ordered phases and phase transitions at high energy and low spatial dimensions. It addresses open questions in the field, such as the nature of the MBL phase transition and the role of interactions in localization. The review concludes that MBL represents a new frontier in quantum statistical mechanics, with potential applications in quantum technologies.