The development of two original Monte Carlo models of solvent diffusion into a polymer is described. Employing a coarse grained model of a polymer solution on a regular lattice, the dynamic properties of both the solvent and polymer molecules can be observed. The "Simple" Monte Carlo model reliably reproduces Case I dynamics, but no departure from this is seen for any reasonable model parameters. This "Simple" Monte Carlo model is unable to reproduce Case II diffusion dynamics. One reason for this is that in this Monte Carlo model the processes of solvent diffusion and polymer relaxation are entirely independent processes. In this thesis it is suggested that a simple Monte Carlo model of this type will always produce Case I diffusion dynamics. The dynamic algorithm described in this work relies on simple instantaneous molecular motions between neighbouring lattice sites. It is shown that a diffusion process based on these motions is purely concentration dependent, relying only on the current state of the system. To use the Monte Carlo method to simulate Case II diffusion dynamics, the diffusion process is made time dependent by incorporating a history dependent model of diffusion first proposed by Crank (CRANK 1953). In this "History Dependent" Monte Carlo model the motions of both the solvent and the polymer are no longer instantaneous, but occur at a rate that approaches equilibrium by a first order process governed by a relaxation time characteristic of the viscoelastic relaxation of the polymer. This "History Dependent" Monte Carlo model successfully simulates most of the features of Case II diffusion and also demonstrates a return to Case I diffusion in the limit of long times. Unlike many models of Case II diffusion, this Monte Carlo model is able to simultaneously model the microscopic motions of both the solvent and the polymer molecules. This novel feature demonstrates the formation of a discontinuous moving boundary between the rubbery polymer and the glassy polymer that is typical of Case II diffusion dynamics.