The global blue economy makes a significant contribution to the world’s economy, providing resources and jobs to approximately 500 million people around the world. Plastic pollution in the oceans is a growing issue due to the slow degradation of plastic items combined with the formation of microplastics. Furthermore, many of these plastics are produced from nonrenewable petrochemicals and many of the items are from single-use packaging. There is growing interest in plastics from renewable resources where many useful polymers are extractable from materials within the marine environment; consequently their use in plastics supports developing a sustainable blue economy. Common marine-sourced polysaccharide polymers include alginates, chitosan, and carrageenan, where chitosan is a waste by-product from the fish and shell-fish industry, so its use contributes to a more circular economy. Additionally, protein polymers available from fish include collagen, gelatin, and myofibrillar, while algae can produce polylactic acid. The development of sustainable packaging materials is important to reduce food waste and promote a more circular economy. This review focuses on research into films derived from marine sources in particular alginate and chitosan for packaging applications, where often other compounds are added such as plasticizers and fillers including natural materials. These bioplastic films often also have functional compounds added to improve their antioxidant or antimicrobial properties. It is important to consider the end-of-life options for any new plastic packaging materials to limit pollution and contribute to a more circular economy.