The concepts and methods of toplogy and geometry have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity and particle physics. This book can be considered an advanced textbook on modern applications and recent developments in this field of research. Written as a set of largely self-contained extensive lectures, the book gives an introduction to toplogical concepts in gauge theories, BRST quantization, chiral anomalies, sypersymmetric solitons and noncommutative geometry. It will be of benefit to postgraduate students, teaching newcomers to the field and lecturers looking for advanced material.