Many previous investigations have indicated that speech data has inherent low-dimensional structure and that it may be possible to efficiently represent speech using only a small number of parameters. This view is motivated by the fact that articulatory movement is limited by physiological constraints and thus the speech production apparatus has only limited degrees of freedom. Also, the set of sounds used in human spoken communication is only a small subset of all producible sounds. A number of dimensionality reduction methods capable of discovering such underlying structure have previously been applied to speech. However, if speech lies on a manifold nonlinearly embedded in high-dimensional space, as has been proposed in the past, classic linear dimensionality reduction methods would be unable to discover this embedding. In this dissertation a number of manifold learning, also referred to as nonlinear dimensionality reduction, methods are applied to speech to explore the possibility of underlying nonlinear manifold structure. This dissertation describes a number of existing manifold learning methods and details the application of these methods to high-dimensional feature representations of speech data. Representations derived from the conventional magnitude spectrum and less widely used phase spectrum are investigated. The manifold learning methods used in this study are locally linear embedding, Isomap, and Laplacian eigenmaps. The classic linear method, principal component analysis (PCA), is also applied to facilitate the comparison of linear and nonlinear methods. The resulting low-dimensional representations are analysed through visualisation, phone recognition, and speaker recognition experiments. The recognition experiments are used as a means of evaluating how much meaningful discriminatory information is contained in the low-dimensional representations produced by each method. These experiments also serve to display the potential value of these methods in speech processing applications. The manifold learning methods are shown to be capable of producing meaningful lowdimensional representations of speech data suggesting speech has low-dimensional manifold structure. In general, these methods are found to outperform PCA in low dimensions, indicating that speech may lie on a manifold nonlinearly embedded in high-dimensional space. Phone classification experiments show that Isomap can offer improvements over standard features and PCA-transformed features. Investigation of magnitude and phase spectrum representations found both to have similar low-dimensional structure and confirm that the phase spectrum contains useful information for phone discrimination. Results indicate that combining magnitude and phase spectrum information yields improvements in phone classification tasks. A method to combine magnitude and phase spectrum features for increased phone classification accuracy without large increases in feature dimensionality is also described.