  We study extended Weyl groups of domestic and wild type. We start with an
extended Coxeter-Dynkin diagram, attach to it a set of roots (vectors) in an
$\mathbb{R}$-space and define the extended Weyl groups as groups that are
generated by the reflections related to these roots, the so called simple
reflections. We relate to such a group $W$ a generalized root system, and
determine the structure of $W$. In particular we present a normal form for the
elements of $W$. Further we define Coxeter transformations in $W$, and show the
transitivity of the Hurwitz action on the set of reduced reflection
factorizations of a Coxeter transformation where the reflections are the
conjugates of the simple reflections in $W$.
  We give two applications of our results. In the context of representation
theory of algebras, we establish an isomorphism between the poset of thick
subcategories that are generated by exceptional sequences of a hereditary
connected ext-finite abelian $k$-category with a tilting object and the poset
of elements in the extended Weyl group that are below a Coxeter transformation
with respect to the absolute order. The second application concerns the theory
of unimodal singularities. In particular, we provide an answer to a question of
Brieskorn for the classical monodromy operator in the case of hyperbolic
singularities.
