In recent years, a field that has risen to prominence within mathematics and engineering is that of PDE-constrained optimization, which has numerous applications within these subject areas. In this talk, we consider the development of preconditioned iterative methods for a number of PDE-constrained optimization problems. In particular, we discuss challenges arising from the interior point solution of problems with additional box constraints, the solution of nonlinear problems arising from chemotaxis, and the application of deferred correction approaches to improve the discretization error obtained in the time variable.