the purpose of this book is to present kinematics of rigid bodies and systems in an axiomatic framework, stressing invariant relations. the audience addressed is engineers, mathematicians and computer scientists working in research and development that involves rigid-body motions. the areas of application are robotics, mechanism design, spacecraft dynamics, computer graphics, solid modelling, finite elements, etc. the most important features of the book are its axiomatic approaches on invariant principles, thus departing from a common but obscure practice of studying general principles based on component-form relations. the book fills the gap between rigid-body kinematics and the theory of kinematic chains, including results published for the first time. the reader can derive the following benefits from this book: a sound theoretical formulation of rigid-body kinematics following an axiomatic presentation, i.e., definitions, theorems and corollaries are given in logical order, the necessary proofs being included after each theorem and corollary. invariant relations between the time rates of change of the rotation invariants and angular velocity and acceleration are derived. these can be useful to those writing computational algorithms for the simulation, control and graphic display of motions of mechanical systems in a great variety of fields.