This thesis describes the mathematical modelling and analysis of liquid crystal systems when an electric field is applied. This analysis is performed for Nematic, Twisted nematic, and Super-twisted nematic cells and for Polymer Gels. The mathematical techniques employed are: linear and non-linear stability analysis; perturbation theory; and the method of matched asymptotic expansions. For conventional nematic systems analytic expressions are obtained which describe the distortion of the liquid crystal and the coupled electric field at low, intermediate, and high applied voltages. Aspects of the dynamics are considered for both strong and weak anchoring and also with the inclusion of a coupled flow. It is shown that certain weakly anchored nematic systems admit travelling wave solutions. This is particularly relevant to the relaxation of polymer gels. For these systems a model is proposed which treats the polymer matrix as comprising thin orientated fibrils which act as weak anchoring sites distributed throughout the liquid crystal. The model is shown to correctly predict many of the experimentally observed properties of polymer gels. Specifically it predicts the enhanced critical voltage for such systems and also indicates that they saturate at a voltage proportionately close to the critical voltage. The model predicts that the decay of a polymer gel from a highly distorted state occurs via a travelling wave. This in turn implies that the decay constant will depend linearly on the width of the cell containing the gel.