  In this paper we answer the question: `what kind of a structure can a general
multicategory be enriched in?' The answer is, in a sense to be made precise,
that a multicategory of one type can be enriched in a multicategory of the type
one level up.
  In the case of ordinary categories this reduces to something surprising: a
category may be enriched in an `fc-multicategory', a very general kind of
2-dimensional structure encompassing monoidal categories, plain
multicategories, bicategories and double categories. (It turns out that
fc-multicategories also provide a natural setting for the bimodules
construction.) An extended application is given: the relaxed multicategories of
Borcherds and Soibelman are explained in terms of enrichment.
