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Neutron stars are very dense stars composed almost entirely of neutrons. As such, they should be able to be described by Quantum Chromodynamics (QCD). As QCD is a very complicated theory from which it is difficult to produce quantitative results we rely on effective theories to describe QCD physics. It has previously been shown that the Skyrme model, which has topological soliton solutions that can be identified as baryons, is such a low energy effective field theory for QCD. In this thesis, after presenting background material in chapters 1, 2 and 3, we explore the results of attempting to use the theory proposed by Skyrme to model neutron stars by investigating two models. The first, discussed in chapter 4 and based on original research, considers rational map ansatz solutions to the Skyrme model. By coupling the model using this ansatz to gravity and introducing a new way of stacking together the shell-like solutions that form we find minimum energy configurations that are stable models of neutron stars. They are, however, slightly too small to be considered a good model so a second approach is tried. The second model considers Skyrme crystal configurations. By using a relation between the energy per baryon of a Skyrme crystal and its anisotropic deformations we are able to find two equations of state for the crystal. These are combined with a Tolman-Oppenheimer-Volkoff equation, generalised to describe anisotropic deformations, to model neutron stars. We find that below a critical mass all deformations are isotropic and above it they are anisotropic up to a particular maximum mass and that this approach compares well with experimental observations. This second model is described in chapter 5 and is based on original research.