Recent experimental and theoretical work in literature has showed that several biological attachment devices are concave in shape and that such a shape enhances adhesion strength. In this paper, an analytical model is developed for the pull-off forces required to separate a rigid axisymmetric concave punch from an elastic half-space with the goal of understanding how concave shapes seen in both biological and man-made adhesive systems can outperform other geometries. Significant enhancement in adhesion strength compared with that of a flat punch was predicted for both spherical and elliptical concave punch profiles. Increasing the degree of the concavity initially increases the resulting pull-off force, but for higher depths the contact interface will separate before pull-off forces higher than those for the flat punch are achieved. Experiments performed with machined aluminum elliptical concave punches on gelatin qualitatively verified the trends predicted by the analytical model, and the maximum pull-off forces measured are within 80% of the predicted optimum pull-off force. These findings emphasize the benefits of concave geometries for designing interfaces which enhance adhesion through geometry alone.