Extraocular muscle pulley bands were described by Tenon in 1805 as "faisceaux tendineux" acting as "poulies de renvoi." The Passive and Active Pulley Hypotheses propose that these connective-tissue bands between muscle and bony orbital rim limit vertical shift of the horizontal rectus muscle belly in up- and downgaze, caused by the muscle's tendency to assume the shortest path from origin to insertion. The band's attachment to the muscle moves 20 mm sagittally when the eye looks from 50° left to 50° right, however, impeding vertical muscle stabilization. Sliding of the muscle in a sleeve would permit sagittal movement, but four anatomical studies could not confirm that. The band would have to be elastic: We measured it after orbital exenteration and found it to be slack, however, and once extended, very stiff. Our research group in Amsterdam suggested in 1984 that the retrobulbar fat and its enveloping connective-tissue sheets including the intermuscular membrane keep muscle bellies in place. We compared horizontal-rectus-muscle positions in up- and down-gaze using frontal CTs through the posterior pole of the eye. The bellies stayed in place while, anteriorly, the tendons bent up- and downward. We also found that the paths of horizontal rectus muscles were curved outwards in horizontal CTs. We surmised that retrobulbar pressure in the fat, resulting from four rectus muscles pulling the eyeball into the orbit, is contained by rectus muscles and connective-tissue sheets and that the resulting tension in the sheets keeps the muscles in place. Years later we repeated the CT study in a Crouzon patient whose bony orbital rim was displaced 2cm posteriorly, preventing pulley-band fixation to the bone: No vertical shift of horizontal rectus muscle bellies occurred in up- and down-gaze. Finally, we developed a mathematical finite-element model of orbit, muscles, fat and eyeball to study whether fat with enveloping connective-tissue sheets could keep eye muscles in place. In simulated eye movements, the retrobulbar fat, with low elasticity as found in vivo, not only kept the eyeball in place but also horizontal rectus muscle bellies in up- and down-gaze and vertical recti in left- and right-gaze.