  A new rigorous approach to conformal field theory is presented. The basic
objects are families of complex-valued amplitudes, which define a meromorphic
conformal field theory (or chiral algebra) and which lead naturally to the
definition of topological vector spaces, between which vertex operators act as
continuous operators. In fact, in order to develop the theory, M\"obius
invariance rather than full conformal invariance is required but it is shown
that every M\"obius theory can be extended to a conformal theory by the
construction of a Virasoro field.
  In this approach, a representation of a conformal field theory is naturally
defined in terms of a family of amplitudes with appropriate analytic
properties. It is shown that these amplitudes can also be derived from a
suitable collection of states in the meromorphic theory. Zhu's algebra then
appears naturally as the algebra of conditions which states defining highest
weight representations must satisfy. The relationship of the representations of
Zhu's algebra to the classification of highest weight representations is
explained.
