  In this thesis we study sets of points in the plane and their Voronoi
diagrams, in particular when the points coincide. We bring together two ways of
studying point sets that have received a lot of attention in recent years:
Voronoi diagrams and compactifications of configuration spaces. We study moving
and colliding points and this enables us to introduce `limit Voronoi diagrams'.
We define several compactifications by considering geometric properties of
pairs and triples of points. In this way we are able to define a smooth, real
version of the Fulton-MacPherson compactification. We show how to define
Voronoi diagrams on elements of these compactifications and describe the
connection with the limit Voronoi diagrams.
