The selective hydrogenation of polyunsaturated fatty acids (PUFAs) and their derivatives to their monounsaturated counterparts is a critical catalytic step in various disciplines, such as biofuels and food chemistry. Furthermore, the necessary transformation of the chemical industry towards sustainability and circularity demands the development of new platform chemicals from renewable resources. The selective hydrogenation of PUFAs offers the flexibility to use a variety of commodity feedstocks in existing processes in the chemical value chain; this is indispensable for staying competitive and in shaping a sustainable future-proof chemical industry. The naturally occurring differences in the composition of vegetable fats and oils, especially the degree of saturation of the carbon chain, necessitate a preparation step to provide a standardized feedstock for existing chemical processes. Since many chemical reactions towards valuable intermediates start from mono-unsaturated substrates, selectivity plays a crucial role in hydrogenation. Furthermore, standardization alleviates the need to grow speciality monocultures or to utilize edible substrates, to further reducing the negative impact on society and the environment. In this critical review, all the relevant and pioneering catalytic transformations throughout history up to this time and place are discussed and evaluated towards their applicability and usefulness for the provision of renewable-based resources for the chemical industry, with particular emphasis on the selectivity towards mono-unsaturated substrates.