  Let V be a simple vertex operator algebra satisfying the following
conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0,
the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is
isomorphic to V as a V-module. (ii) Every weak V-module gradable by nonnegative
integers is completely reducible. (iii) V is C_2-cofinite. We announce a proof
of the Verlinde conjecture for V, that is, of the statement that the matrices
formed by the fusion rules among irreducible V-modules are diagonalized by the
matrix given by the action of the modular transformation \tau\mapsto -1/\tau on
the space of characters of irreducible V-modules. We discuss some consequences
of the Verlinde conjecture, including the Verlinde formula for the fusion
rules, a formula for the matrix given by the action of \tau\mapsto -1/\tau and
the symmetry of this matrix. We also announce a proof of the rigidity and
nondegeneracy property of the braided tensor category structure on the category
of V-modules when V satisfies in addition the condition that irreducible
V-modules not equivalent to V has no nonzero elements of weight 0. In
particular, the category of V-modules has a natural structure of modular tensor
category.
