|Abstract:|| Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Typically, Total variation (TV) regularisation is used due to its edge-preserving and smoothing properties. In this talk, we focus on the use of PDEs for the optimal design of several Imaging models. Starting from the use of higher-order and non-smooth variants of classical heat-type diffusion models often employed to counteract well-known TV reconstruction drawbacks (such as staircasing), we focus in particular on the use of non-smooth optimisation approaches with PDE constraints in the context of optimal parameter estimation for some non-standard image denoising problems. Finally, we consider some TV-based models recently reinterpreted in the framework of graphs and applied to some image segmentation problems.