An analytical model based on the Johnson-Kendall-Roberts (JKR) theory of adhesion was used to study the contact mechanics and adhesion of periodically rough surfaces. The relation of the applied load to the contact area and the work of adhesion W was found in closed form for arbitrary surface profiles. Our analysis showed that when the parameter alpha = 2/pi beta root 2W rho /E* > alpha* [where alpha* is a numerical constant of order, one beta one is the aspect ratio of a typical surface profile (or asperity), and rho is the number of asperities per unit length], the surfaces will jump into contact with each other with no applied load, and the contact area will continue to expand until the two surfaces are in full. contact. The theory was then extended to the non-JKR regime in which the region where the surface forces act is no longer confined to a small region near the contact zone. Exact solution was also obtained for this case. An exact analysis of the effect of entrapped air on the mechanics of adhesion and contact was also enacted. The results showed that interaction between asperities should be taken into consideration in contact-mechanics models of adhesion or friction.