The ideal food packaging materials are recyclable, biodegradable, and compostable. Starch from plant sources, such as tubers, legumes, cereals, and agro-industrial plant residues, is considered one of the most suitable biopolymers for producing biodegradable films due to its natural abundance and low cost. The chemical modification of starch makes it possible to produce films with better technological properties by changing the functional groups into starch. Using biopolymers extracted from agro-industrial waste can add value to a raw material that would otherwise be discarded. The recent COVID-19 pandemic has driven a rise in demand for single-use plastics, intensifying pressure on this already out-of-control issue. This review provides an overview of biopolymers, with a particular focus on starch, to develop sustainable materials for food packaging. This study summarizes the methods and provides a potential approach to starch modification for improving the mechanical and barrier properties of starch-based films. This review also updates some trends pointed out by the food packaging sector in the last years, considering the impacts of the COVID-19 pandemic. Perspectives to achieve more sustainable food packaging toward a more circular economy are drawn.