  The subject of this paper is Beurling's celebrated extension of the Riemann
mapping theorem \cite{Beu53}. Our point of departure is the observation that
the only known proof of the Beurling-Riemann mapping theorem contains a number
of gaps which seem inherent in Beurling's geometric and approximative approach.
We provide a complete proof of the Beurling-Riemann mapping theorem by
combining Beurling's geometric method with a number of new analytic tools,
notably $H^p$-space techniques and methods from the theory of
Riemann-Hilbert-Poincar\'e problems. One additional advantage of this approach
is that it leads to an extension of the Beurling-Riemann mapping theorem for
analytic maps with prescribed branching. Moreover, it allows a complete
description of the boundary regularity of solutions in the (generalized)
Beurling-Riemann mapping theorem extending earlier results that have been
obtained by PDE techniques. We finally consider the question of uniqueness in
the extended Beurling-Riemann mapping theorem.
