Dynkin diagrams appear in various mathematical
contexts like the theory of simple Lie algebras, finite reflection
groups, representation theory, or surface singularities. One of
the connexions between these different fields is made explicit
by the theorem of Grothendieck-Brieskorn-Slodowy. It describes a
relation between the singularities associated to finite subgroups
of $\operatorname{SU}(2)$, that were studied by Felix Klein, and the geometry of
simple Lie algebras. In the talk I will illustrate this theorem in
detailed examples and discuss a far reaching generalisation
via the notion of Poisson deformations. This last part of the
talk reports on joint work with Christoph Sorger and Namikawa
Yoshinori.