Teaching mathematically gifted students is a challenging pursuit, especially at the secondary level. However, characterizing teachers who can successfully meet this challenge has so far been scarcely addressed directly. In this paper, elements of the desired profile of such teachers are extracted from the general literature on mathematical giftedness. The difficulty to recruit and train teachers to satisfy this profile, resulting from the reality of severe shortages of mathematics teachers in many countries, is described. Findings are presented from a study on a unique 2-year training course for teachers of mathematically gifted and advanced students, including analyses of the prerequisites of the course, the syllabus and the profile of graduates. The findings point to a problem where compromises in entry requirements, made in light of shortage in suitable candidates, may limit the number of successful graduates even in comprehensive and carefully designed programs. Implications of these results for decision makers are discussed.