To meet consumers’ demand for natural foods, edible oil producers and food processing industries are searching for alternatives to synthetic antioxidants to protect oils against oxidation. Antioxidant compounds extracted from different plant parts (e.g., flowers, leaves, roots, and seeds) or sourced from agri-food industries, including residues left after food processing, attract consumers for their health properties and natural origins. This review, starting from a literature research analysis, highlights the role of natural antioxidants in the protection of edible oils against oxidation, with an emphasis on the emerging and sustainable strategies to preserve oils against oxidative damage. Sustainability and health are the main concerns of food processing industries. In this context, the aim of this review is to highlight the emerging strategies for the enrichment of edible oils with biomolecules or extracts recovered from plant sources. The use of extracts obtained from vegetable wastes and by-products and the blending with oils extracted from various oil-bearing seeds is also pointed out as a sustainable approach. The safety concerns linked to the use of natural antioxidants for human health are also discussed. This review, using a multidisciplinary approach, provides an updated overview of the chemical, technological, sustainability, and safety aspects linked to oil protection.