Anthocyanins are the main plant pigments responsible for the color of flowers, fruits, and vegetative organs of many plants, and are applied also as safe food colorants. They are efficient antioxidants. In this review, the reactivity of anthocyanins and their aglycones, anthocyanidins, in the main antioxidant assays, and their reactions with reactive oxygen and nitrogen species, effects of interactions with other compounds and metal ions on the antioxidant activity of anthocyanins and the electrochemical properties of anthocyanins are presented. Numerous cases of attenuation of oxidative stress at the cellular and organismal levels by anthocyanins are cited. The direct and indirect antioxidant action of anthocyanins, the question of the specificity of anthocyanin action in complex extracts, as well as limitations of cellular in vitro assays and biomarkers used for the detection of antioxidant effects of anthocyanins, are critically discussed.