  This survey is about combinatorial objects related to reflection groups and
their applications in representation theory and arithmetic geometry. Coxeter
groups and folded galleries in Coxeter complexes are introduced in detail and
illustrated by examples. Further it is explained how they relate to retractions
in Bruhat-Tits buildings and to the geometry of affine flag varieties and
affine Grassmannians. The goal is to make these topics accessible to a wide
audience.
