The introduction of hybrid materials in regenerative medicine has solved some problems related to the mechanical and bioactive properties of biomaterials. Calcium phosphates and their derivatives have provided the basis for inorganic components, thanks to their good bioactivity, especially in bone regeneration. When mixed with biodegradable polymers, the result is a synergic association that mimics the composition of many tissues of the human body and, additionally, exhibits suitable mechanical properties. Together with the development of nanotechnology and new synthesis methods, hybrids offer a promising option for the development of a third or fourth generation of smart biomaterials and scaffolds to guide the regeneration of natural tissues, with an optimum efficiency/cost ratio. Their potential bioactivity, as well as other valuable features of hybrids, open promising new pathways for their use in bone regeneration and other tissue repair therapies. This review provides a comprehensive overview of the different hybrid organic-inorganic scaffolding biomaterials developed so far for regenerative therapies, especially in bone. It also looks at the potential for research into hybrid materials for other, softer tissues, which is still at an initial stage, but with very promising results.