Gauge theories and noncommutative geometry

Afriendly introduction to (part of) noncommutative differential geometry is given with emphasis on, and applications to, examples coming from gauge theories. In particular, starting from (what physicists call) monopoles and instantons as connections on bundles over spheres, we arrive at very natural deformations of spaces and bundles. The `noncommutative' manifolds and vector bundles that one obtains have very interesting and rich geometrical structures that can be described with natural tools.