  Convergence and analytic extension are of fundamental importance in the
mathematical construction and study of conformal field theory. We review some
main convergence results, conjectures and problems in the construction and
study of conformal field theories using the representation theory of vertex
operator algebras. We also review the related analytic extension results,
conjectures and problems. We discuss the convergence and analytic extensions of
products of intertwining operators (chiral conformal fields) and of $q$-traces
and pseudo-$q$-traces of products of intertwining operators. We also discuss
the convergence results related to the sewing operation and the determinant
line bundle and a higher-genus convergence result. We then explain conjectures
and problems on the convergence and analytic extensions in orbifold conformal
field theory and in the cohomology theory of vertex operator algebras.
