Interactive deformation via control handles is essential in computer graphics for the modeling of 3D geometry. Deformation control structures include lattices for free-form deformation and skeletons for character articulation, but this report focuses on cage-based deformation. Cages for deformation control are coarse polygonal meshes that encase the to-be-deformed geometry, enabling high-resolution deformation. Cage-based deformation enables users to quickly manipulate 3D geometry by deforming the cage. Due to their utility, cage-based deformation techniques increasingly appear in many geometry modeling applications. For this reason, the computer graphics community has invested a great deal of effort in the past decade and beyond into improving automatic cage generation and cage-based deformation. Recent advances have significantly extended the practical capabilities of cage-based deformation methods. As a result, there is a large body of research on cage-based deformation. In this report, we provide a comprehensive overview of the current state of the art in cage-based deformation of 3D geometry. We discuss current methods in terms of deformation quality, practicality, and precomputation demands. In addition, we highlight potential future research directions that overcome current issues and extend the set of practical applications. In conjunction with this survey, we publish an application to unify the most relevant deformation methods. Our report is intended for computer graphics researchers, developers of interactive geometry modeling applications, and 3D modeling and character animation artists.