In this chapter we begin by finding the conformal algebra in a Minkowski space-time, that is, the set of transformations that leave the metric invariant up to a scale factor. In dimensions greater than two the conformal algebra is finite dimensional, but in two dimensions it is infinite-dimensional. We will find that if a field theory possesses conformal symmetry, then its energy-momentum tensor is traceless. For a two-dimensional classical theory conformal invariance implies the theory has an infinite number of conserved quantities, which are moments of the energy-momentum tensor.

In two-dimensional quantum theories one finds that the conformal algebra becomes modified by a central term associated with the required normal ordering. The central term contains a constant, which is called the central charge, upon which the nature of the representations of the conformal algebra crucially depends. As for any symmetry of a quantum field theory, the conformal symmetry of a two-dimensional quantum theory implies certain Ward identities. However, only a finite-dimensional sub-algebra of the infinite-dimensional conformal algebra is globally defined on the appropriate space-time and we will find that the Green's functions are only invariant under these transformations. Nonetheless, for a certain class of theories, called the minimal models, which have special central charges we can use the Ward identities corresponding to all the conformal transformations to place constraints on the Green's functions, which actually determine them.