  The functorial mathematical definition of conformal field theory was first
formulated approximately 30 years ago. The underlying geometric category is
based on the moduli space of Riemann surfaces with parametrized boundary
components and the sewing operation. We survey the recent and careful study of
these objects, which has led to significant connections with quasiconformal
Teichmuller theory and geometric function theory.
  In particular we propose that the natural analytic setting for conformal
field theory is the moduli space of Riemann surfaces with so-called
Weil-Petersson class parametrizations. A collection of rigorous analytic
results is advanced here as evidence. This class of parametrizations has the
required regularity for CFT on one hand, and on the other hand are natural and
of interest in their own right in geometric function theory.
