The flow around bluff bodies is determined by flow separations that result in a vortex street in their wake. Depending on the shape, the flow topology and the time dependent behaviour of the separation process can be strongly influenced by the Reynolds number, even if the bodies have sharp edges. The reason for this dependence is that the state of the boundary layer has a far-reaching influence on the entire flow field and its topological structure. In particular, the location of the laminar/turbulent transition in the boundary layer, or in the separated shear layer, is an important parameter. In general, changes in the topological structure of the separated flow can lead to dramatic changes in the steady and unsteady aerodynamic forces, acting on the body. The basic mechanisms, including their sensitivity to small perturbations, will be discussed using examples from flow around a circular and a rectangular cylinder as generic two-dimensional bluff bodies. It is well known that the Reynolds number dependence plays a dominant role for the circular cylinder; however the flow around a rectangular cylinder is often seen as Reynolds number independent, because of the sharp edges. This statement loses general validity when the separated flow has the possibility of reattachment, forming separation bubbles. The discussions of the basic phenomena are based on a series of experiments carried out over a wide range of Reynolds numbers from 104 up to 107 in the high-pressure wind tunnel in Göttingen. In addition to the circular and rectangular cylinders, a trapezoidal bridge section and a thick airfoil at a high attack angle were investigated. In all cases, appropriately shifting the value of Reynolds number led to dramatic changes in the force coefficients and the Strouhal number. Oil-flow pictures were used to make the changes in the topological structure of the separated flow visible. The changes in the flow are triggered by the laminar/turbulent transition and its location, and have characteristics that appear to be universally valid because they occur in similar form in flow over different bodies. Thus we can assume that a deeper insight will help to better understand flow separation effects in general. Although the presentation is focused on Reynolds number effects in incompressible flow, the influence of the Mach number will be examined, using the example of a circular cylinder. Finally the propensity of bluff bodies to flow-induced vibrations will be discussed and demonstrated by simple table experiments.