Differential Geometry and Mathematical Physics : Part II. Fibre Bundles Topology and Gauge Fields Fibre bundles and connections Linear connections and Riemannian geometry Homotopy theory of principal fibre bundles. Classification Cohomology theory of fibre bundles. Characteristic classes Clifford algebras spin structures and Dirac operators The Yang Mills equation Matter fields and model building The gauge orbit space Elements of quantum gauge theory A Field restriction and field extension B The Conformal Group of the 4 sphere C Simple Lie algebras. Root diagrams D z function regularization E K theory and index bundles F Determinant line bundles G Eilenberg MacLane spaces References. Index. The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks: Geometry and topology of fibre bundles Clifford algebras spin structures and Dirac operators Gauge theory. Written in the style of a mathematical textbook it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory. The first building block includes a number of specific topics like invariant connections universal connections H structures and the Postnikov approximation of classifying spaces. Given the great importance of Dirac operators in gauge theory a complete proof of the Atiyah Singer Index Theorem is presented. The gauge theory part contains the study of Yang Mills equations (including the theory of instantons and the classical stability analysis) the discussion of various models with matter fields (including magnetic monopoles the Seiberg Witten model and dimensional reduction) and the investigation of the structure of the gauge orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem anomalies and the implementation of the non generic gauge orbit strata in the framework of Hamiltonian lattice gauge theory. The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from a graduate level.