This research aimed to examine the views of preservice mathematics teachers (PMTs) on mathematically giftedness. Case study, one of the qualitative research design, was used and the study involved 25 sophomore mathematics teaching undergraduate students from Turkey. Data were collected with a semi-structured interview form with open-ended questions and analyzed with content analysis method. As a result of the analysis of the data three themes comprising of definitions of giftedness, characteristics of gifted students including sub-themes of cognitive, affective and social traits, and mathematically gifted students including sub-themes of cognitive and affective traits were emerged. As a result of the study it was determined that the majority of the participants thought that the gifted students have exceptional performance in intelligence or creativity in a specific area. The cognitive traits of gifted students were favorably listed as being curious or asking questions, strong verbal skills, high performance compared to the peers, fast thinking and learning, creativity and abstract thinking. The affective traits mainly included sensitivity, sense of justice, impatience, sense of humour, intense interest/concentration, diligence and patience, responsibility, curiosity, perfectionism and motivation. The social characteristics of gifted were emerged as preference to communicate with adults, enhanced social relationship (resulting in leadership), rarity in society and valued by society. On the other hand, characteristics of mathematically gifted students have been classified as cognitive and affective traits according to the views of PMTs. Cognitive traits of mathematically gifted students mostly composed of quick, unusual and multiple solutions to problems, analytical thinking, proficiency in mathematics, interest in numbers at an early age, and questioning. Affective characteristics of mathematically gifted students have been found to be introversion, peer disapproval, self-confidence, responding maturely, good sense of self-expression, sense of wonderment, and willingness to learn. As a result, most of the views of PMTs were found to be similar to the literature. Considering that the participants took elective course "Teaching Mathematics to Gifted Students", it can be stated that it is important to give lectures on giftedness to prospective teachers in their undergraduate education.