What is a horocyclic product, and how is it related to lamplighters ?

This is a rather personal introductory outline of an interesting class of geometric,
resp. graph- & group-theoretical structures. After an introductive section about
their genesis, the general construction of horocyclic products is presented. Three
closely related basic structures of this type are explained in more detail: Diestel-
Leader graphs, treebolic spaces, and Sol-groups, resp. -manifolds. Emphasis is
on their geometry, isometry groups, quasi-isometry classification and boundary
at infinity. Subsequently, it is clarified under which parametrisation they admit
discrete groups of isometries acting with compact quotient. Finally, further de-
velpoments are reviewed briefly.