Conformal field theory can be defined using the associativity and the commutativity of the product of quantum fields (operator product expansion). An important difference between conformal field theory and classical commutative associative algebra is "the divergence" arising from the product of quantum fields, a difficulty that appears in quantum field theory in general. In this talk we will explain that in the two-dimensional case this algebra can be controlled by the representation category of a vertex operator algebra and that the convergence of quantum fields is described by the operad structure of the configuration space.