We develop the theory of Hopf bimodules for a finite rigid tensor category C.
Then we use this theory to define a distinguished invertible object D of C and
an isomorphism of tensor functors ?^{**} and D tensor ^{**}? tensor D^{-1}.
This provides a categorical generalization of D. Radford's S^4-formula for
finite dimensional Hopf algebras and its generalizations for weak Hopf algebras
and for quasi-Hopf algebras, and conjectured in general in \cite{EO}. When C is
braided, we establish a connection between the above isomorphism and the
Drinfeld isomorphism of C. We also show that a factorizable braided tensor
category is unimodular (i.e., D=1). Finally, we apply our theory to prove that
the pivotalization of a fusion category is spherical, and give a purely
algebraic characterization of exact module categories.