The presence of irrelevant features in training data is a significant obstacle for many machine learning tasks, since it can decrease accuracy, make it harder to understand the learned model and increase computational and memory requirements. One approach to this problem is to extract appropriate features. General approaches such as Principal Components Analysis (PCA) are successful for a variety of applications, however they can be improved upon by targeting feature extraction towards more specific problems. More recent work has been more focused and considers sparser formulations which potentially have improved generalisation. However, sparsity is not always efficiently implemented and frequently requires complex optimisation routines. Furthermore, one often does not have a direct control on the sparsity of the solution. In this thesis, we address some of these problems, first by proposing a general framework for feature extraction which possesses a number of useful properties. The framework is based on Partial Least Squares (PLS), and one can choose a user defined criterion to compute projection directions. It draws together a number of existing results and provides additional insights into several popular feature extraction methods. More specific feature extraction is considered for three objectives: matrix approximation, supervised feature extraction and learning the semantics of two-viewed data. Computational and memory efficiency is prioritised, as well as sparsity in a direct manner and simple implementations. For the matrix approximation case, an analysis of different orthogonalisation methods is presented in terms of the optimal choice of projection direction. The analysis results in a new derivation for Kernel Feature Analysis (KFA) and the formation of two novel matrix approximation methods based on PLS. In the supervised case, we apply the general feature extraction framework to derive two new methods based on maximising covariance and alignment respectively. Finally, we outline a novel sparse variant of Kernel Canonical Correlation Analysis (KCCA) which approximates a cardinality constrained optimisation. This method, as well as a variant which performs feature selection in one view, is applied to an enzyme function prediction case study.