  We introduce the Frobenius-Schur indicator for categories with duality to
give a category-theoretical understanding of various generalizations of the
Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras,
weak Hopf C*-algebras and association schemes. Our framework also clarifies a
mechanism how the `twisted' theory arises from the ordinary case. As a
demonstration, we give a twisted Frobenius-Schur theorem for semisimple
quasi-Hopf algebras. We also give several applications to the quantum SL_2.
