This book discusses the different facets of the problem of compressing data pertaining to whereabouts-in-time information for mobile entities. It gives a comprehensive overview of the state of the art in terms of existing techniques as well as the impact of various contexts associated with modeling and representing the motion. The first part of this book presents a global overview of the problem of data compression in general and throughout the history, illustrates the different categorizations of compression approaches, and positions the rest of the book in these settings. It discusses separately the facets of compressing spatial data (polylines, cartography, and beyond) and temporal data (temporal databases, time series, streaming data). The second part of this book explores in detail the various issues arising when compression is attempted in the realm of moving objects management, both for point-objects and evolving shapes. It starts with discussing the basic settings and the related solutions and fundamental techniques common to various application and analyzes the benefits and trade-offs associated with mobile data reduction in both online and (near) real-time settings. It also covers the impact of the different distance functions used to capture the quality of the compression process. Subsequently, it incorporates the role of different contexts such as energy issues when tracking in wireless sensor networks, known restrictions of motion (e.g., road networks), etc. Other key topics range from the role of data compression in clustering mobile data to the impact of various semantics-based features (such as symbolic trajectories and warehousing of spatio-temporal data). Compression of Mobility Data concludes with a discussion of the possible future research directions associated with different aspects of compressing spatio-temporal data.