  We introduce a combinatorial theory of horospherical stacks which is
motivated by the work of Geraschenko and Satriano on toric stacks. A
horospherical stack corresponds to a combinatorial object called a stacky
coloured fan. We give many concrete examples, including a class of easy-to-draw
examples called coloured fantastacks. The main results in this paper are
combinatorial descriptions of horospherical stacks, the morphisms between them,
their decolourations, and their good moduli spaces.
