This book introduces the concepts of tensor algebras and differentiable manifolds to the intermediate-level student. It describes analytical and geometrical structures built on these basic concepts. Those structures -- which include differential forms and their integration, flows, Lie derivatives, distributions and their integrability conditions, connections, and pseudo-Riemannian and symplectic manifolds -- are then applied to the description of the fundamental ideas and Hamiltonian and Lagrangian mechanics, and special and general relativity. This book is designed to be accessible to the mathematics or physics student with a good standard undergraduate background, who is interested in obtaining a broader perspective of the rich interplay of mathematics and physics before deciding on a specialty.