We present a calculation of the attractive (van ver Waals) forces that are to be expected between tips and samples in experiments using atomic-force microscopes and scanning tunneling microscopes, and we discuss the important roles that these attractive forces, and their repulsive counterparts, play in the experiments. It is shown that the attractive forces have been underestimated in previous work, and we discuss important consequences of this fact. For example, we conclude that atomic-force microscopes should be operated with their tips touching the surface, but with their cantilevers deflected towards the surface so that the associated spring forces counteract most of the attractive forces. Further, we suggest that these microscopes may operate more satisfactorily in liquid environments because of the resulting reduction in the attractive forces.