  We consider C-graded vertex algebras, which are vertex algebras V with a
C-grading such that V is an admissible V-module generated by 'lowest weight
vectors'. We show that such vertex algebras have a 'good' representation theory
in the sense that there is a Zhu algebra A(V) and a bijection between simple
admissible V-modules and simple A(V)-modules. We also consider pseudo vertex
operator algebras, which are C-graded vertex algebras with a conformal vector
such that the homogeneous subspaces of V are generalized eigenspaces for L(0);
essentially, these are VOAs that lack any semisimplicity or integrality
assumptions on L(0). As a motivating example, we show that deformation of the
conformal structure (conformal flow) of a strongly regular VOA (eg a lattice
theory, or WZW model) is a path in a space whose points are PVOAs.
