  We introduce the notion of finite right (respectively left) numerical index
on a bimodule $X$ over C*-algebras A and B with a bi-Hilbertian structure. This
notion is based on a Pimsner-Popa type inequality. The right (respectively
left) index element of X can be constructed in the centre of the enveloping von
Neumann algebra of A (respectively B). X is called of finite right index if the
right index element lies in the multiplier algebra of A. In this case we can
perform the Jones basic construction. Furthermore the C*--algebra of bimodule
mappings with a right adjoint is a continuous field of finite dimensional
C*-algebras over the spectrum of Z(M(A)), whose fiber dimensions are bounded
above by the index. We show that if A is unital, the right index element
belongs to A if and only if X is finitely generated as a right module.
  We show that bi-Hilbertian, finite (right and left) index C*-bimodules are
precisely those objects of the tensor 2-C*-category of right Hilbertian
C*-bimodules with a conjugate object, in the sense of Longo and Roberts, in the
same category.
