Knowledge about the own pose is key for all mobile robot applications. Thus pose estimation is part of the core functionalities of mobile robots. Over the last two decades, LiDAR scanners have become the standard sensor for robot localization and mapping. This article aims to provide an overview of recent progress and advancements in LiDAR-based global localization. We begin by formulating the problem and exploring the application scope. We then present a review of the methodology, including recent advancements in several topics, such as maps, descriptor extraction, and cross-robot localization. The contents of the article are organized under three themes. The first theme concerns the combination of global place retrieval and local pose estimation. The second theme is upgrading single-shot measurements to sequential ones for sequential global localization. Finally, the third theme focuses on extending single-robot global localization to cross-robot localization in multi-robot systems. We conclude the survey with a discussion of open challenges and promising directions in global LiDAR localization. To our best knowledge, this is the first comprehensive survey on global LiDAR localization for mobile robots.