Organic farming is a comprehensive production management system that fosters and improves the health of agroecosystems, encompassing biodiversity, biological cycles, and soil biological activity. The potato (Solanum tuberosum L.) is a crucial crop in organic farming systems, standing out as one of the most highly demanded organic products on the market. Among all crops, with potatoes, there is a very large yield gap between organic and conventional systems, attributable mainly to its intensive nutrient demands. The present review, considering the most relevant scientific literature worldwide, discusses the contemporary state of knowledge on crop nutrition and soil fertility management in organic potato crop production, analyzing the effects of animal manures, green manures, organic amendments, and biostimulants on organic potato tuber yield and quality. Overall, the main findings show a particular combination is needed to effectively maintain good soil fertility, satisfy the nutritional needs of the crop, and overcome the difference in potato yield between organic and conventional farming methods while meeting consumer demand. This combination entails using an animal manure or leguminous green manure with an organic soil amendment, and even better with a biofertilizer, such as a mycorrhizae-fungus-based one. It also emerged that more targeted studies are needed to select appropriate cultivars for organic potato farming systems to optimize this environmentally friendly production method.