We have two different axiomatic approaches to chiral conformal field theory (CFT). The conformal net approach is based on the theory of operator algebras on Hilbert spaces (C^*-algebras and von Neumann algebras) and it is the chiral CFT version of algebraic quantum field theory (AQFT). On the other hand, the vertex operator algebra (VOA) approach is based on an algebraic reformulation of the relevant properties of conformal invariant quantum fields on the circle.