The development of new scaffolds and materials for tissue engineering is a wide and open realm of material science. Among solutions, the use of biopolymers represents a particularly interesting area of study due to their great chemical complexity that enables creation of specific molecular architectures. However, biopolymers do not exhibit the properties required for direct application in tissue repair—such as mechanical and electrical properties—but they do show very attractive chemical functionalities which are difficult to produce through in vitro synthesis. The combination of biopolymers with nanostructured carbon fillers could represent a robust solution to enhance composite properties, producing composites with new and unique features, particularly relating to electronic conduction. In this paper, we provide a review of the field of carbonaceous nanostructure-containing biopolymer composites, limiting our investigation to tissue-engineering applications, and providing a complete overview of the recent and most outstanding achievements.