  One of the basic geometric objects in conformal field theory (CFT) is the the
moduli space of Riemann surfaces whose $n$ boundaries are ''rigged'' with
analytic parametrizations. The fundamental operation is the sewing of such
surfaces using the parametrizations to identify points. An alternative model is
the moduli space of $n$-punctured Riemann surfaces together with local
biholomorphic coordinates at the punctures. We refer to both of these moduli
spaces as the "rigged Riemann moduli space".
  By generalizing to quasisymmetric boundary parametrizations, and defining
rigged Teichmueller spaces in both the border and puncture pictures, we prove
the following results: (1) The Teichmueller space of a genus-$g$ surface
bordered by $n$ closed curves covers the rigged Riemann and rigged Teichmueller
moduli spaces of surfaces of the same type, and induces complex manifold
structures on them. (2) With this complex structure the sewing operation is
holomorphic. (3) The border and puncture pictures of the rigged moduli and
rigged Teichmueller spaces are biholomorphically equivalent.
  These results are necessary in rigorously defining CFT (in the sense of G.
Segal), as well as for the construction of CFT from vertex operator algebras.
