Plants are exposed to a variety of abiotic and biotic stresses leading to increased formation of reactive oxygen species (ROS) in plant cells. ROS are capable of oxidizing proteins, pigments, lipids, nucleic acids, and other cell molecules, disrupting their functional activity. During the process of evolution, numerous antioxidant systems were formed in plants, including antioxidant enzymes and low molecular weight non-enzymatic antioxidants. Antioxidant systems perform neutralization of ROS and therefore prevent oxidative damage of cell components. In the present review, we focus on the biosynthesis of non-enzymatic antioxidants in higher plants cells such as ascorbic acid (vitamin C), glutathione, flavonoids, isoprenoids, carotenoids, tocopherol (vitamin E), ubiquinone, and plastoquinone. Their functioning and their reactivity with respect to individual ROS will be described. This review is also devoted to the modern genetic engineering methods, which are widely used to change the quantitative and qualitative content of the non-enzymatic antioxidants in cultivated plants. These methods allow various plant lines with given properties to be obtained in a rather short time. The most successful approaches for plant transgenesis and plant genome editing for the enhancement of biosynthesis and the content of these antioxidants are discussed.