Fresh vegetables and fruits have always been the mainstays of good nutrition as providers of fiber, beneficial phytochemicals (such as vitamins and phenolic compounds), and minerals. Today and in the future, biofortification is a promising strategy to increase the concentration of these compounds. Considering the importance of minerals in human health, the enrichment of fresh produce for consumption has been considered through specific agronomic approaches. This review discusses, in detail, the latest findings on vegetable agronomic biofortification, aimed at increasing the concentration of crucial minerals, such as iron (Fe), zinc (Zn), iodine (I), selenium (Se), molybdenum (Mo), and silicon (Si), in edible portions, focusing on the direct and indirect effects of this strategy. Although agronomic biofortification is considered a feasible technique, the approach is complex due to the many interactions between the microelement bioavailability for both plants and consumers. Therefore, the effects of biofortification on human health and the influence of beneficial and antinutritional compounds were discussed in detail to analyze the advantages and disadvantages of this practice.