The following is now available, at http://www.dpmms.cam.ac.uk/~leinster/ Generalized Enrichment for Categories and Multicategories In this paper we answer the question: `what kind of a structure can a general multicategory be enriched in?' (Here `general multicategory' is used in the sense of the author, Burroni or Hermida.) 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 particular, we will be able to speak of a T_n-multicategory enriched in a T_{n+1}-multicategory, where T_n is the monad expressing the pasting-together of n-opetopes. The answer for general multicategories reduces to something surprising in the case of ordinary categories: 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. We also explore enrichment for some multicategories other than just categories. An extended application is given: the relaxed multicategories of Borcherds and Soibelman are explained in terms of enrichment. Tom Leinster PS - There's been the odd problem in the past with the web address; if it doesn't work, try substituting "can" for "www", or send me an email.