Tissue engineering may be defined as the application of biological, chemical and engineering principles toward the repair, restoration or regeneration of living tissue using biomaterials, cells and biologically active molecules alone or in combinations. The rapid restoration of tissue biomechanical function represents a great challenge, highlighting the need to mimic tissue structure and mechanical behavior through scaffold designs. For this reason, several biodegradable and bioresorbable materials, as well as technologies and scaffold designs, have been widely investigated from an experimental and/or clinical point of view. Accordingly, this review aims at stressing the importance of polymer-based composite materials to make multifunctional scaffolds for tissue engineering, with a special focus on bone, ligaments, meniscus and cartilage. Moreover, polymer-based nanocomposites will also be briefly introduced as an interesting strategy to improve the biological and mechanical performances of polymer scaffolds, especially for bone tissue engineering.