The paper discusses the phenomenon of mathematical giftedness, especially manifested at early stages of life of future outstanding mathematicians, taken in its socio-cultural context. The authors suggest that the images of mathematical giftedness are formed differently in various cultural contexts and thus can imply different settings of the educational institutions that can accordingly ignore, encourage, or restrain the students considered gifted. The paper focuses on the cases of traditional mathematics in several Asian countries (China, Vietnam, and Japan) and of modem mathematics in Soviet Union/Russia in order to provide examples of different patterns of forming the image of mathematical giftedness and of the corresponding educational approaches.