The control of fluid flow through a channel or the de-noising and segmentation of medical imagesor applications of data assimilation result in optimisation problems involving PDEs as state equations. Of burgeoning interest is the control of problems involving free boundaries and interfaces. Inverse problems for differential equation are increasingly important in applications. Apart from the use of PDE optimisation, key challenges include the formulation of these problems on function space, study of the resulting posterior probability measure in singular limits of interest, such as small observational noise, and the development of effective sampling methodologies, often MCMC based, to probe the posterior measure. Modelling and solution require tools across [A, C, P, S].