Linear dimensionality reduction techniques, notably principal component analysis, are widely used in climate data analysis as a means to aid in the interpretation of datasets of high dimensionality. These hnear methods may not be appropriate for the analysis of data arising from nonlinear processes occurring in the climate system. Numerous techniques for nonlinear dimensionality reduction have been developed recently that may provide a potentially useful tool for the identification of low-dimensional manifolds in climate data sets arising from nonlinear dynamics. In this thesis I apply three such techniques to the study of El Niño/Southern Oscillation variability in tropical Pacific sea surface temperatures and thermocline depth, comparing observational data with simulations from coupled atmosphere-ocean general circulation models from the CMIP3 multi-model ensemble.