This textbook presents the mathematics that is foundational to the theory of quantum fields, featuring a rigorous survey of selected methods from algebra, manifolds, fiber bundles, Lie groups and representation theory. Only undergraduate-level knowledge of mathematics is required. It addresses beginning graduate students in mathematics and physics alike who seek a rigorous approach to modern quantum physics. Of course, the growth and complexity of modern viewpoints and techniques used in quantum field theory require to concentrate on basic ideas. Choices have to be made. The authors follow a chronological path from classical mechanics in Chapter 1 to the Standard Model of particle physics in Chapter 9. They present the basic mathematics of quantum mechanics including the \(C^*\) algebra approach and the path integral quantization in the sense of Feynman. After a short exposition of classical field theory, the quantization of free fields is explained in a rigorous manner. The quantization of interacting fields, however, is only touched upon, while subjects like perturbation theory, scattering and renormalization are discussed in some detail. Having read this book, students will feel encouraged to seek further information and deeper insight by using the quoted literature.