This detailed yet accessible text provides an essential introduction to the advanced mathematical methods at the core of theoretical physics.The book steadily develops the key concepts required for an understanding of symmetry principles and topological structures, such as group theory, differentiable manifolds, Riemannian geometry, and Lie algebras.Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering.The material has been subject to more than twenty years of feedback from students, ensuring that explanations and examples are lucid and considered, and numerous worked examples and exercises reinforce key concepts and further strengthen readers' understanding.This text unites a wide variety of important topics that are often scattered across different books, and provides a solid platform for more specialized study or research.