Factorization of the operator product algebra in conformal field theory into independent left and right components is investigated. For those theories in which factorization holds we propose an ansatz for the number of independent amplitudes which appear in the fusion rules, in terms of the crossing matrices of conformal blocks in the plane. This is proved to be equivalent to a recent conjecture by Verlinde. The monodromy properties of the conformal blocks of 2-point functions on the torus are investigated. The analysis of their short-distance singularitities leads to a precise definition of Verlinde's operations.