This thesis presents elliptical rolling contact joints as an alternative to circular rolling contact and conventional revolute joints where high quality force transmission "low friction and backlash" with variable output are desired. Parameters specific to the joint and its position are developed in terms of relative link angles and elliptical surface geometry. These parameters are used to generate the basic forward kinematics for elliptical rolling link toggle mechanisms with oscillatory motion and high mechanical advantage. As large compressive loads are characteristic of such mechanisms, stress conditions are identified and principles for joint stability with variable, precision outputs are discussed. Finally, application is made to self-adjusting passive force closures with a case study of the MUSCLE Brake (Multi-toggle Self-adjusting Connecting-Linked Electromechanical) disc brake caliper. Elliptical rolling contact joints are shown to offer several benefits over circular rolling contact, including: reduced Hertz contact stresses and flexure bending stresses, variable output velocity, maximum use of contact interface by distributing small rotations across surfaces of small curvature, reduced forces on stabilizing members, increased mechanical advantage due to eccentricity, and no-slip pure rolling provided exclusively by connecting links (or flexures) without the need for gear teeth or friction.