  The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a
simple example of a moduli space and also a nice (crepant) resolution of
singularities of the symmetric power $S^{(n)}$. For many phenomena expected for
moduli spaces and nice resolutions of singular varieties it is a model case.
Hilbert schemes of points have connections to several fields of mathematics,
including moduli spaces of sheaves, Donaldson invariants, enumerative geometry
of curves, infinite dimensional Lie algebras and vertex algebras and also to
theoretical physics. This talk will try to give an overview over these
connections.
