  This lecture consists of two sections. In section 1 we consider the simplest
version of a q-deformed Heisenberg algebra as an example of a noncommutative
structure. We first derive a calculus entirely based on the algebra and then
formulate laws of physics based on this calculus. Then we realize that an
interpretation of these laws is only possible if we study representations of
the algebra and adopt the quantum mechanical scheme. It turns out that
observables like position or momentum have discrete eigenvalues and thus space
gets a lattice-like structure.
  In section 2 we study a framework for higher dimensional noncommutative
spaces based on quantum groups. The Poincare-Birkhoff-Witt property and
conjugation properties play an essential role there. In these spaces
derivatives are introduced and based on these derivatives a q-deformed
Heisenberg algebra can be constructed.
