This book is intended to provide a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles, and Chern forms that are essential for a deeper understanding of both classical and modem physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, elasticity theory, the geometry and topology of Kirchhoff 's electric circuit laws, soap films, special and general relativity, the Dirac operator and spinors, and gauge fields including Yang-Mills, the Aharonov-Bohm effect, Berry phase, and instanton winding numbers. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space ; consequently, the book should be of interest also to mathematics students. This book will be useful to graduate and advanced undergraduate students of physics, engineering, and mathematics. It can be used as a course text or for self-study.