This open access book presents a new approach to kinematics and dynamics of a broad spectrum of mechanical systems that is based on modern analytical (differential geometry) and computational (differential equation) solution methods that enable treatment of challenging applications that conventional methods are unable to reliably simulate. In addition to classical ideal holonomic multibody models, nonholonomic systems and systems encountering complex friction effects are treated. No background in abstract topology of differential geometry is assumed of the reader. Mathematical and computational concepts are introduced using only multivariable calculus at the undergraduate level applied to model practical engineering applications encountered in the field. Examples are treated throughout to provide a foundation for understanding modern concepts and techniques that have heretofore only been presented at an abstract mathematical level in theoretical literature that is not generally accessible to the practicing engineer. MATLAB computer code is provided that implements numerical methods presented in the book.