Biopolymer-based packaging materials have become of greater interest to the world due to their biodegradability, renewability, and biocompatibility. In recent years, numerous biopolymers—such as starch, chitosan, carrageenan, polylactic acid, etc.—have been investigated for their potential application in food packaging. Reinforcement agents such as nanofillers and active agents improve the properties of the biopolymers, making them suitable for active and intelligent packaging. Some of the packaging materials, e.g., cellulose, starch, polylactic acid, and polybutylene adipate terephthalate, are currently used in the packaging industry. The trend of using biopolymers in the packaging industry has increased immensely; therefore, many legislations have been approved by various organizations. This review article describes various challenges and possible solutions associated with food packaging materials. It covers a wide range of biopolymers used in food packaging and the limitations of using them in their pure form. Finally, a SWOT analysis is presented for biopolymers, and the future trends are discussed. Biopolymers are eco-friendly, biodegradable, nontoxic, renewable, and biocompatible alternatives to synthetic packaging materials. Research shows that biopolymer-based packaging materials are of great essence in combined form, and further studies are needed for them to be used as an alternative packaging material.