Food packaging plays a vital role in protecting food products from environmental damage and preventing contamination from microorganisms. Conventional food packaging made of plastics produced from unrenewable fossil resources is hard to degrade and poses a negative impact on environmental sustainability. Natural biopolymers are attracting interest for reducing environmental problems to achieve a sustainable society, because of their abundance, biocompatibility, biodegradability, chemical stability, and non-toxicity. Active packaging systems composed of these biopolymers and biopolymer-based composites go beyond simply acting as a barrier to maintain food quality. This review provides a comprehensive overview of natural biopolymer materials used as matrices for food packaging. The antioxidant, water barrier, and oxygen barrier properties of these composites are compared and discussed. Furthermore, biopolymer-based composites integrated with antimicrobial agents—such as inorganic nanostructures and natural products—are reviewed, and the related mechanisms are discussed in terms of antimicrobial function. In summary, composites used for active food packaging systems can inhibit microbial growth and maintain food quality.