D-differentiation is a unified operation that enables aspects of differential geometry to be developed and presented from a new perspective. This book is the first comprehensive and self-contained treatment of this new method. It demonstrates, concisely but without sacrificing rigour or intelligibility, how even elementary concepts in differential geometry can be reformulated to obtain new and valuable insights. In addition, D-differentiation has applications in several areas of physics, such as classical mechanics, solid-state physics and general relativity. This book will prove useful to all users of D-differentiation - from advanced graduate students onwards - and to those researching into new approaches to some branches of physics and mathematics.