Different monolithic preconditioning techniques for the incompressible Navier-Stokes equations are introduced and compared to a selection of block preconditioners. In particular, two-level additive overlapping Schwarz methods are used to set up monolithic preconditioners and to approximate the inverses that arise in the block preconditioners. To construct the second level GDSW-type (generalized Dryja–Smith–Wildund) coarse spaces are used. These highly-scalable, parallel GDSW preconditioners have been implemented in the solver framework FROSch (Fast and Robust Overlapping Schwarz), which is part of the software library Trilinos. Monolithic preconditioners are robust because they account for the coupling terms in the system matrix on both levels, that is, in the local and coarse problems. In comparison, block preconditioners, mostly based on block-diagonal and block-triangular preconditioners, such as the PCD (pressure convection-diffusion) and SIMPLE (semi-implicit method for pressure linked equations) preconditioners, often yield higher iteration counts while having a lower setup cost compared to monolithic approaches. In this talk, the parallel performance of the different preconditioning methods for incompressible fluid flow problems is investigated and compared using a finite element implementation based on the FEDDLib (Finite Element and Domain Decomposition Library) and the overlapping Schwarz preconditioners from the Trilinos package FROSch. Furthermore, the robustness of these methods is tested for a range of Reynolds numbers with respect to a realistic problem setting.