Let Σ be a Riemann surface with n distinguished points p 1,..., p n . We prove that the set of n-tuples (φ1,..., φ n ) of univalent mappings φ i from the unit disc $$ \mathbb{D} $$ into Σ mapping 0 to p i , with non-overlapping images and quasiconformal extensions to a neighbourhood of $$ \overline {\mathbb{D}} $$ , carries a natural complex Banach manifold structure. This complex structure is locally modeled on the n-fold product of a two complex-dimensional extension of the universal Teichmuller space. Our results are motivated by Teichmuller theory and two-dimensional conformal field theory.