Tissue engineering and regenerative medicine has been providing exciting technologies for the development of functional substitutes aimed to repair and regenerate damaged tissues and organs. Inspired by the hierarchical nature of bone, nanostructured biomaterials are gaining a singular attention for tissue engineering, owing their ability to promote cell adhesion and proliferation, and hence new bone growth, compared with conventional microsized materials. Of particular interest are nanocomposites involving biopolymeric matrices and bioactive nanosized fillers. Biodegradability, high mechanical strength, and osteointegration and formation of ligamentous tissue are properties required for such materials. Biopolymers are advantageous due to their similarities with extracellular matrices, specific degradation rates, and good biological performance. By its turn, calcium phosphates possess favorable osteoconductivity, resorbability, and biocompatibility. Herein, an overview on the available natural polymer/calcium phosphate nanocomposite materials, their design, and properties is presented. Scaffolds, hydrogels, and fibers as biomimetic strategies for tissue engineering, and processing methodologies are described. The specific biological properties of the nanocomposites, as well as their interaction with cells, including the use of bioactive molecules, are highlighted. Nanocomposites in vivo studies using animal models are also reviewed and discussed.