Robots interact with the world through curves and surfaces — the subjects of study in differential geometry. This book applies the moving-frame method, developed extensively by Élie Cartan, and the adjoint approach, conceived by Ernesto Cesàro, to study the kinematics of two surfaces subject to rolling contact and sliding-rolling contact to demonstrate the applications in robotic in-hand manipulation.Firstly, it explores two surfaces, and the geometry of both surfaces comes into play. Secondly, the book focuses on the geometry of the two surfaces within the encompassing space (extrinsic) rather than within the surfaces (intrinsic) because the book is concerned with the kinematics of one surface in three-dimensional Euclidean space — the real world. The book then concludes by applying this approach in robotic in-hand manipulation in the last chapter.