Collagen, gelatin, silk fibroin, hyaluronic acid, chitosan, alginate, and cellulose are biocompatible and non-cytotoxic, being attractive natural polymers for medical devices for both soft and hard tissues. However, such natural polymers have low bioactivity and poor mechanical properties, which limit their applications. To tackle these drawbacks, collagen, gelatin, silk fibroin, hyaluronic acid, chitosan, alginate, and cellulose can be combined with bioactive glass (BG) nanoparticles and microparticles to produce composites. The incorporation of BGs improves the mechanical properties of the final system as well as its bioactivity and regenerative potential. Indeed, several studies have demonstrated that polymer/BG composites may improve angiogenesis, neo-vascularization, cells adhesion, and proliferation. This review presents the state of the art and future perspectives of collagen, gelatin, silk fibroin, hyaluronic acid, chitosan, alginate, and cellulose matrices combined with BG particles to develop composites such as scaffolds, injectable fillers, membranes, hydrogels, and coatings. Emphasis is devoted to the biological potentialities of these hybrid systems, which look rather promising toward a wide spectrum of applications.