Introduction to non-commutative geometry and its applications in physics

In order to explore the foundations of physics at very short distances, a mathematical framework describing a quantized version of space-time geometry and quantum fields thereon is desired. One possible road to take is to introduce a non-commutative space-time and to formulate quantum field theories within this new framework. Apart from this fundamental motivation, there are other more accessible physical situations which may be described by an effective non-commutative space, such as e.g. the Quantum Hall effect. In this talk I will introduce the concept of non-commuative geometry and how to formulate field theories in such a space. I will then focus mostly on issues of renormalizability, ultraviolet/infrared mixing and prospects of such models as theories of all fundamental interactions.