Nanoscale biomaterials have garnered immense interest in the scientific community in the recent decade. This review specifically focuses on the application of three nanomaterials, i.e., graphene and its derivatives (graphene oxide, reduced graphene oxide), carbon nanotubes (CNTs) and nanocellulose (cellulose nanocrystals or CNCs and cellulose nanofibers or CNFs), in regenerating different types of tissues, including skin, cartilage, nerve, muscle and bone. Their excellent inherent (and tunable) physical, chemical, mechanical, electrical, thermal and optical properties make them suitable for a wide range of biomedical applications, including but not limited to diagnostics, therapeutics, biosensing, bioimaging, drug and gene delivery, tissue engineering and regenerative medicine. A state-of-the-art literature review of composite tissue scaffolds fabricated using these nanomaterials is provided, including the unique physicochemical properties and mechanisms that induce cell adhesion, growth, and differentiation into specific tissues. In addition, in vitro and in vivo cytotoxic effects and biodegradation behavior of these nanomaterials are presented. We also discuss challenges and gaps that still exist and need to be addressed in future research before clinical translation of these promising nanomaterials can be realized in a safe, efficacious, and economical manner.