This textbook explores the theory behind differentiable manifolds and investigates various physics applications along the way. Basic concepts, such as differentiable manifolds, differentiable mappings, tangent vectors, vector fields, and differential forms, are briefly introduced in the first three chapters. Chapter 4 gives a concise introduction to differential geometry needed in subsequent chapters. Chapters 5 and 6 provide interesting applications to connections and Riemannian manifolds. Lie groups and Hamiltonian mechanics are closely examined in the last two chapters. Included throughout the book are a collection of exercises of varying degrees of difficulty. Differentiable Manifolds is intended for graduate students and researchers interested in a theoretical physics approach to the subject. Prerequisites include multivariable calculus, linear algebra, differential equations, and a basic knowledge of analytical mechanics.