  Our purpose is to pursue the rigorous construction of Liouville Quantum Field
Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two
authors in the context of the Riemann sphere and inspired by the 1981 seminal
work by Polyakov. In this paper, we investigate the case of simply connected
domains with boundary. We also make precise conjectures about the relationship
of this theory to scaling limits of random planar maps with boundary
conformally embedded onto the disk.
