Detailed finite element calculations are carried out in order to study the mechanical response of a compliant layer sandwiched between a rigid cylindrical Aat punch and a rigid substrate. Two cases of practical interest are considered: one in which the layer is perfectly bonded to the punch and the substrate and one in which the interface between the punch and the layer is frictionless. The substrate is assumed to be perfectly bonded to the adhesive layer in both cases. Analytic expressions are obtained for the stresses away from the edges, and the effect of lateral constraint is examined. The compliances of the loading systems for both cases are obtained numerically, and accurate analytic expressions are determined based on these numeric results. The nature of the stress fields near the contact edge are explored, and their connections with the energy release rate are determined. The relevance of these calculations to two recent adhesion tests is discussed.