Palm oil is widely used in the food industry because of its lower cost, high oxidative stability index, long shelf-life, and a reasonable replacement of trans fats. However, increased palm oil production reduces biodiversity, damages the ecosystem, and poses health risks to humans. Unsustainable development of palm plantations has caused deforestation and loss of natural habitat, rendering many species (Sumatran orangutans, elephants, and tigers) critically endangered. Similarly, decomposition and burning of carbon-rich soil in vast and deep peatlands is increasing carbon emissions. Furthermore, excessive consumption of palmitic acid (and other saturated fats except stearic acid) increases bad cholesterol and the risk of cardiovascular diseases. Therefore, we need healthier, stable, and solid/semisolid oils at room temperature with longer shelf-life and without trans fats. Here, we review the advancements in the development of sunflower oil varieties containing high stearic (∼18%) and high oleic (∼70%) acids which makes them healthy and sustainable alternatives to palm oil. First, the high-stearic-high-oleic sunflower crops can have grain and oil yield as high as 4036 and 1685 kg/ha and oleic and stearic acid yield up to ∼73 and ∼21%. Second, high-stearic-high-oleic oils obtained from mutant and hybrid sunflower cultivars have higher oxidative stability index and therefore have better stability, quality, and functionality than regular sunflower oil. For example, the oxidative stability index of commercially available Nutrisun at 110°C is six times greater than that of regular sunflower oil. Finally, recent advances have made several mutant and hybrid cultivars with high grain and oil yield and high levels of stearic and oleic acids available. Given this progress, natural healthy high-stearic-high-oleic sunflower oil can now be grown in both the hemispheres in a sustainable manner with the currently available advanced technology and without damaging the ecosystem as is currently happening with palm oil cultivation.