In this work three dimensionless parameters are introduced in the debate concerning hard contact models. These parameters are related to the overall adhesive contact area, curved surface contribution and surface interaction forces outside the contact region. With the variations of these three parameters, the relations and transitions between the different hard contact models such as Hertz, Bradley, Johnson-Kendall-Roberts (JKR), Derjaguin-Muller-Toporov (DMT) and Maugis-Dugdale (MD) models are presented in a systematic way. The combination of the three parameters provides a new hybrid model. The influence of these three parameters on the contact between spheres has been studied. By analyzing the pressure profiles of contact region, two new instability jumps are proposed. The instability jumps together with the three parameters are used to explain some recent experimental and numerical observations which deviate more or less from those predicated by the classical hard contact models. (C) Koninklijke Brill NV, Leiden, 2011