The development of biomaterials for the restoration of the normal tissue structure–function relationship in pathological conditions as well as acute and chronic injury is an area of intense investigation. More recently, the use of tailored or composite hydrogels for tissue engineering and regenerative medicine has sought to bridge the gap between natural tissues and applied biomaterials more clearly. By applying traditional concepts in engineering composites, these hydrogels represent hierarchical structured materials that translate more closely the key guiding principles required for improved recovery of tissue architecture and functional behavior, including physical, mass transport, and biological properties. For tissue-engineering scaffolds in general, and more specifically in composite hydrogel materials, each of these properties provide unique qualities that are essential for proper augmentation and repair following disease and injury. The broad focus of this review is on physical properties in particular, static and dynamic mechanical properties provided by composite hydrogel materials and their link to native tissue architecture and, ultimately, tissue-specific applications for composite hydrogels.