In the last decade, biopolymer matrices reinforced with nanofillers have attracted great research efforts thanks to the synergistic characteristics derived from the combination of these two components. In this framework, this review focuses on the fundamental principles and recent progress in the field of aliphatic polyester-based nanocomposites for regenerative medicine applications. Traditional and emerging polymer nanocomposites are described in terms of polymer matrix properties and synthesis methods, used nanofillers, and nanocomposite processing and properties. Special attention has been paid to the most recent nanocomposite systems developed by combining alternative copolymerization strategies with specific nanoparticles. Thermal, electrical, biodegradation, and surface properties have been illustrated and correlated with the nanoparticle kind, content, and shape. Finally, cell-polymer (nanocomposite) interactions have been described by reviewing analysis methodologies such as primary and stem cell viability, adhesion, morphology, and differentiation processes.