The topics of tracking moving objects and moving shapes have been extensively researched in multiple communities - from Moving Objects Databases (MOD) and spatio-temporal data management, through image/video processing and traffic management, to environmental and ecology studies. This paper gives a summary of the topics discussed in the advanced seminar on tracking objects and shapes, as well as an overview of its proposed structure. After a brief introduction and motivation-survey of different research fields and societal applications, the first part of the seminar will give a historic survey of the fundamental techniques for tracking mobile objects. The second part will give an overview of the approaches popular in MOD and spatiotemporal data management communities (tracking and querying, streaming data, map-matching, etc.). The third part is the central one - discussing the issues and solutions in distributed tracking of moving objects and shapes: From topological predicates and trends detection, through tracking deformable shapes, to specifics of indoor tracking. The fourth major part is intended to be a 'potpourri-style' review of different application contexts and the popular approaches for tracking individual objects and shapes - spanning from collective motion analysis in social networks and animal herds, through toxic elements, pollutants, and geoprocesses (landslides), to different approaches for visual analytics in this context. The main objective of this advanced seminar is to provide a cohesive overview of the different perspectives on motion tracking; the corresponding approaches for its effective management; and possibilities for other research directions.