Food packaging plays an imperative role in the food processing sector by safeguarding foods from their point of harvesting until the moment of consumption. In recent years, biopolymers have attracted the attention of the scientific community as an alternative to conventional packaging materials. Among the available biopolymer sources, a lot of the focus has been on polysaccharides due to their superior barrier properties against gases, oils, and odors and their processing versatility. Moreover, there is also a growing interest in aliphatic polyester as a potential replacement for petrochemical-based synthetic plastics. Both polysaccharides and aliphatic polyesters have gained popularity in sustainable food packaging due to their unique characteristics, including their low cost, availability, biodegradability, gas and moisture barrier properties, film-forming capabilities, excellent heat resistance, and ability to be processed into films, trays, and coatings. This review highlights the structural features, properties, and recent advancements of several vital polysaccharides, namely, starch, chitosan, cellulose, alginate, pectin, carrageenan, and aliphatic polyesters, including polylactic acid (PLA) and polyhydroxybutyrate (PHB) for developing packaging materials, and their applications in the food industry. Conventional packaging and future perspectives of biopolymer-based food packaging are also comprehensively covered in this review.