We are used to describing physical phenomena by continuous and commuting space-time coordinates. At very short distances we run into difficulties with divergencies in a quantum field theory, for instance divergent masses. As early as 1930 Werner Heisenberg proposed a solution to this problem by introducing uncertainty relations for space coordinates as well. This then leads directly to the concept of noncommutative coordinates.
Various examples of noncommutative space-time coordinates will be introduced. It will then be shown that quantum field theories and especially gauge theories can be built on such an algebraic system. Some of the examples are also motivated by string theory. The purpose of this talk will be to demonstrate that physical systems can be constructed on noncommutative algebras as well and that they have very specific consequences that can be tested experimentally.