The construction of meaningful observables in models of quantum gravity is a highly non-trivial task, but necessary in order to study their continuum physics. In this thesis several such observables are identified in lattice models of quantum gravity. In dynamical triangulations in two dimensions with topology of the torus we have studied numerically the lengths of non-contractible loops and the moduli parametrizing the conformal structure. Good agreement is found with analytical results from Liouville gravity. Using similar methods we have studied the time evolution of the conformal structure in causal dynamical triangulations in 2+1 dimensions with spatial topology of the torus. An effective model is constructed describing the outcome of the computer simulations and compared to various continuum descriptions of quantum gravity in 2+1 dimensions.