  An introduction is given to some selected aspects of noncommutative geometry.
Simple examples in this context are provided by finite sets and lattices. As an
application, it is explained how the nonlinear Toda lattice and a discrete time
version of it can be understood as generalized sigma-models based on
noncommutative geometries. In particular, in this way one achieves a simple
understanding of the complete integrability of the Toda lattice. Furthermore,
generalized metric structures on finite sets and lattices are briefly
discussed.
