A framework of increasing complexity is developed to describe the mechanics of adhesion and its reverse, de-adhesion or separation, during indentation contact cycles. The importance of the indentation probe stiffness in determination of system stability is emphasized and shown to control the details of the adhesive snap-on instability for a simple logarithmic probe-surface interaction potential and the snap-on and pull-off instabilities (and their existence) for a Mie interaction potential. The work of adhesion is shown to depend critically on the probe stiffness. Indentation contact-mechanics coordinates are introduced and are used to develop the Johnson-Kendall-Roberts (JKR) contact model. The similarities, differences, and conveniences of the logarithmic, Mie, and JKR adhesive indentation contact models are discussed.