Structuring liquid oils has become an active area of research in the past decade, mainly due to pressures to reduce saturated fat intake and eliminate trans fats from our diets. However, replacing hard fats with liquid oil can lead to major changes in the quality of food products. Recent strategies to impart solid-fat functionality to liquid oils include the addition of unusual compounds to oil, leading to its gelation. These include small-molecule organogelators such as phytosterols and 12-hydroxystearic acid, which self-assemble into crystalline fibers which trap oil. Other crystalline additives include waxes, ceramides, monoacylglycerides, and other surfactants. Recently, the polymer ethyl cellulose was reported to form a polymer gel in triacylglyceride (TAG) oils. Other non-conventional strategies also include the formation of protein-stabilized cellular solids with oil trapped within the cells. In this review, we summarize the research on each one of these components in order to provide a comprehensive overview of the state of the area in oleogel research and provide future perspectives.