  fc-multicategories are a very general kind of two-dimensional structure,
encompassing bicategories, monoidal categories, double categories and ordinary
multicategories. We define them and explain how they provide a natural setting
for two familiar categorical ideas. The first is the bimodules construction,
traditionally carried out on suitably cocomplete bicategories but perhaps more
naturally carried out on fc-multicategories. The second is enrichment: there is
a theory of categories enriched in an fc-multicategory, extending the usual
theory of enrichment in a monoidal category. We finish by indicating how this
work is just the simplest case of a much larger phenomenon.
