Developing bio-based and biodegradable materials has become important to meet current market demands, government regulations, and environmental concerns. The packaging industry, particularly for food and beverages, is known to be the world’s largest consumer of plastics. Therefore, the demand for sustainable alternatives in this area is needed to meet the industry’s requirements. This review presents the most commonly used bio-based and biodegradable packaging materials, bio-polyesters, and polysaccharide-based polymers. At the same time, a major problem in food packaging is presented: fungal growth and, consequently, food spoilage. Different types of antifungal compounds, both natural and synthetic, are explained in terms of structure and mechanism of action. The main uses of these antifungal compounds and their degree of effectiveness are detailed. State-of-the-art studies have shown a clear trend of increasing studies on incorporating antifungals in biodegradable materials since 2000. The bibliometric networks showed studies on active packaging, biodegradable polymers, films, antimicrobial and antifungal activities, essential oils, starch and polysaccharides, nanocomposites, and nanoparticles. The combination of the development of bio-based and biodegradable materials with the ability to control fungal growth promotes both sustainability and the innovative enhancement of the packaging sector.