Horospherical stacks and stacky coloured fans

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