  I investigate two discrete models of random geometries, namely simplicial
quantum gravity and quantum string theory. In four-dimensional simplicial
quantum gravity, I show that the addition of matter gauge fields to the model
is capable of changing its phase structure by replacing the branched polymers
of the pure gravity model with a new phase that has a negative string
susceptibility exponent and a fractal dimension of four. Some of the results
are derived from a strong coupling expansion of the model, a technique which is
used here for the first time in this context. In quantum string theory, I study
a discrete version of the IIB superstring. I show that the divergences
encountered in the discretization of the bosonic string are eliminated in the
supersymmetric case. I give theoretical arguments for the appearance of
one-dimensional structures in the region of large system extents that manifest
as a power-law tail in the link length distribution; this is confirmed by
numerical simulations of the model. I also examine a lower-dimensional version
of the IKKT matrix model, in which a similar effect can be observed.
