  In some bicategories, the 1-cells are `morphisms' between the 0-cells, such
as functors between categories, but in others they are `objects' over the
0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many
bicategorical notions do not work well in these cases, because the `morphisms
between 0-cells', such as ring homomorphisms, are missing. We can include them
by using a pseudo double category, but usually these morphisms also induce base
change functors acting on the 1-cells. We avoid complicated coherence problems
by describing base change `nonalgebraically', using categorical fibrations. The
resulting `framed bicategories' assemble into 2-categories, with attendant
notions of equivalence, adjunction, and so on which are more appropriate for
our examples than are the usual bicategorical ones.
  We then describe two ways to construct framed bicategories. One is an
analogue of rings and bimodules which starts from one framed bicategory and
builds another. The other starts from a `monoidal fibration', meaning a
parametrized family of monoidal categories, and produces an analogue of the
framed bicategory of spans. Combining the two, we obtain a construction which
includes both enriched and internal categories as special cases.
