This book provides an introduction to the mathematics of modern physics, presenting concepts and techniques in mathematical physics at a level suitable for advanced undergraduates and beginning graduate students. The aim is to introduce the reader to modern mathematical thinking within a physics setting. Topics covered include tensor/exterior algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book includes exercises and worked examples, to test the students' understanding of the various concepts, as well as extending the themes covered in the main text. The book will be suitable for mathematical and theoretical physicists as well as applied mathematicians.