This book presents the mathematical theory of design for articulated systems called linkages. Robot manipulators, walking machines, and mechanical hands are examples of these systems, all of which rely on simple mechanical constraints to provide a complex workspace for an end-effector. The emphasis of this text is on linkage systems with fewer degrees of freedom than that of a typical robot arm and, therefore, more constraints. The focus is on sizing these constraints to guide the end-effector through a set of task positions. Formulated in this way the design problem is purely geometric in character. The theory is developed for planar linkages before moving to devices that constrain spatial rotation and general spatial displacement. This allows intuition developed from plane geometry to provide insight into the geometry of points and lines in space. This book will be useful to mathematics, engineering, and computer science departments teaching courses on geometric design, geometric modeling, and computer-aided design, as well as the theory of design for mechanical systems and robots.