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The solution of PDE-constrained optimization problems has many important applications, for example in medical imaging, in inverse problems and in the optimization of industrial processes. We solve such problems using an all-at-once approach, which requires solving a linear or nonlinear system that is many times larger than the underlying PDE model. Finding the solution to such a large system is very computationally intensive, requiring iterative linear solvers and specially designed preconditioners. In this project we are speeding up the solution of these problems by using advanced high-performance parallel computers and innovative parallle algorithms, allowing us to reach solutions more quickly, to get more accuracy, and to solve larger problems of the kind that may arise in realistic applications.