The accumulation of synthetic plastics used in packaging applications in landfills and the environment is a serious problem. This challenge is driving research efforts to develop biodegradable, compostable, or recyclable barrier materials derived from renewable sources. Cellulose, chitin/chitosan, and their combinations are versatile biobased packaging materials because of their diverse biological properties (biocompatibility, biodegradability, antimicrobial properties, antioxidant activity, non-toxicity, and less immunogenic compared to protein), superior physical properties (high surface area, good barrier properties, and mechanical properties), and they can be assembled into different forms and shapes (powders, fibers, films, beads, sponges, gels, and solutions). They can be either assembled into packaging films or used as fillers to improve the properties of other biobased polymers. Methods such as preparation of composites, multilayer coating, and alignment control are used to further improve their barrier, mechanical properties, and ameliorate their moisture sensitivity. With the growing application of cellulose and chitin-based packaging materials, their biodegradability and recyclability are also discussed in this review paper. The future trends of these biobased materials in packaging applications and the possibility of gradually replacing petroleum-based plastics are analyzed in the “Conclusions” section.