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This is a rather personal introductory outline of an interesting class ofgeometric, resp. graph- and group-theoretical structures. After an introductivesection about their genesis, the general construction of horocyclic products ispresented. Three closely related basic structures of this type are explained inmore detail: Diestel-Leader graphs, treebolic spaces, and Sol-groups, resp.-manifolds. Emphasis is on their geometry, isometry groups, quasi-isometryclassification and boundary at infinity. Subsequently, it is clarified underwhich parametrisation they admit discrete groups of isometries acting withcompact quotient. Finally, further develpoments are reviewed briefly.  Remark: the posted version contains a correction of a misprint that appearedon the bottom of page 22 in the version which appeared in IMN.

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