A variety of natural biological tissues (e.g., skin, ligaments, spider silk, blood vessel) exhibit 'J-shaped' stress-strain behavior, thereby combining soft, compliant mechanics and large levels of stretchability, with a natural 'strain-limiting' mechanism to prevent damage from excessive strain. Synthetic materials with similar stress-strain behaviors have potential utility in many promising applications, such as tissue engineering (to reproduce the nonlinear mechanical properties of real biological tissues) and biomedical devices (to enable natural, comfortable integration of stretchable electronics with biological tissues/organs). Recent advances in this field encompass developments of novel material/structure concepts, fabrication approaches, and unique device applications. This review highlights five representative strategies, including designs that involve open network, wavy and wrinkled morphologies, helical layouts, kirigami and origami constructs, and textile formats. Discussions focus on the underlying ideas, the fabrication/assembly routes, and the microstructure-property relationships that are essential for optimization of the desired 'J-shaped' stress-strain responses. Demonstration applications provide examples of the use of these designs in deformable electronics and biomedical devices that offer soft, compliant mechanics but with inherent robustness against damage from excessive deformation. We conclude with some perspectives on challenges and opportunities for future research.