Multigrid is one of the most efficient algorithms to solve large linear systems arising from PDE`s and can be used as a key component for the optimal control of the time-dependent Navier--Stokes equations.

This talk presents the basic ideas and highlights some results of the current development of this solver technique. The underlying KKT system is discretised in a monolithic way on the whole space-time domain using finite elements in space and a one-step-$\theta$-scheme in time. A global Newton solver is applied to solve for the nonlinearity. Exploiting the elliptic character of the linearised KKT system in space and time, a space-time multigrid solver is used for the linear subproblems. The resulting solver combination is robust with respect to the considered flow configurations and shows a convergence behaviour which is is quite independent of the number of unknowns in the discrete problem.