An overview of benchmark computations for 2D and 3D laminar flows around a cylinder is given, which have been defined for a comparison of different solution approaches for the incompressible Navier-Stokes eqautions developed within the Priority Research Programme. The exact definition of the benchmarks are recapitulated and the numerical schemes and computers employed by the various participating groups are summarized. A detailed evaluation of the results provided is given, also including a comparison with a reference experiment. The principal purpose of the benchmarks is discussed and some general conclusions which can be drawn from the results are formulated.