Deployable Euler spiral connectors (DESCs) are introduced as compliant deployable flexures that can span gaps between segments in a mechanism and then lay flat when under strain in a stowed position. This paper presents models of Euler spiral beams combined in series and parallel that can be used to design compact compliant mechanisms. Constraints on the flexure parameters of DESCs are also presented. Analytic models developed for the force-deflection behavior and stress were compared to finite element analysis and experimental data. A spinal implant and a linear ratcheting system are presented as illustrative applications of DESCs.