This thesis examines the problem of variational image decomposition, as first introduced and developed by Meyer in [1], and its applications to textured image discrimination and textured image denoising. Image decomposition generally refers to the splitting of an image f into the sum of two or more components, e.g. u, a cartoon component, and v, a texture component. After a brief overview of the use of partial differential equations in image processing, which has become very widespread in recent years, a novel image decomposition model called Improved Edge Segregation, based on the pioneering work of Vese and Osher [2], is put forward, which gives better decomposition results, as it better separates cartoon and texture edges into their proper components. Decomposition with Improved Edge Segregation is generally performed in less time than that from Vese and Osher's model, and gives superior (better quality and faster) texture discrimination results when used in conjunction with Active Contours without Edges [3]. Then a new model, called the Simultaneous Decomposition/Discrimination model, which simultaneously decomposes and segments a textured image is described, also improving decomposition and discrimination quality. Extensions to the image decomposition model of Osher, Solé and Vese [4], which is itself based on the H⁻¹-norm, are subsequently put forward. One such extension decorrelates the cartoon and texture components by using an energy term based on the correlation coefficient between them in local windows, giving better decomposition results. The other combines the decomposition and nonlinear diffusion frameworks, for the purposes of ameliorating denoising performance. First, Perona and Malik [5] nonlinear diffusion is incorporated into decomposition, and subsequently the framework of image denoising with Oriented Laplacians is incorporated. Finally, two models where texture is represented by one subcomponent are presented, one which requires precomputation of image orientations, Orientation-Adaptive Decomposition, and another, Eikonal Orientation-Adaptive Decomposition, which does not. Orientation-Adaptive Decomposition is applied to both the decomposition and the denoising of oriented textures, and some theoretical discussion of Eikonal Orientation-Adaptive Decomposition included. Finally some conclusions and suggestions for future work are given based on the research presented in the thesis.