  This thesis is devoted to the application of random matrix theory to the
study of random surfaces, both discrete and continuous; special emphasis is
placed on surface boundaries and the associated boundary conditions in this
formalism. In particular, using a multi-matrix integral with permutation
symmetry, we are able to calculate the partition function of the Potts model on
a random planar lattice with various boundary conditions imposed. We proceed to
investigate the correspondence between the critical points in the phase diagram
of this model and two-dimensional Liouville theory coupled to conformal field
theories with global $\mathcal{W}$-symmetry. In this context, each boundary
condition can be interpreted as the description of a brane in a family of
bosonic string backgrounds. This investigation suggests that a spectrum of
initially distinct boundary conditions of a given system may become degenerate
when the latter is placed on a random surface of bounded genus, effectively
leaving a smaller set of independent boundary conditions. This curious and
much-debated feature is then further scrutinised by considering the double
scaling limit of a two-matrix integral. For this model, we can show explicitly
how this apparent degeneracy is in fact resolved by accounting for
contributions invisible in string perturbation theory. Altogether, these
developments provide novel descriptions of hitherto unexplored boundary
conditions as well as new insights into the non-perturbative physics of
boundaries and branes.
