The aim of the article is to emphasis the results of theoretical analysis of the problem of abilities and their development in psychological-pedagogical research works. The beginning of scientific research of abilities dates back to the XIX century and is connected with the names of F. Galton, J. Cattel, A. Binet and others. In the beginning of the XX century various theories of intellect appear: bifactor model of intellect by C. Spearman, multifactor theory by L. Thurstone, hierarchical model of intellect by R. Cattel. The counterbalance of multifactor concepts make up one-dimensional models (H. Eysenck). Three most known system theories of abilities in the modern times are a bio-ecological model by S. Cesi, the theory of multiple intellects by H. Gardner and the theory of successful intellect by R. Sternberg. In the second half of the XX century in the foreign psychology biological and cognitive approaches begin to develop. In the domestic science before the beginning of the XX century the research of abilities was at the same level with world science. From the point of view of uncontrolled and massive borrowing and use of «western» concepts of abilities, in 1936 by the decision of the government there was put a ban and scientific researches were stopped. Fundamental developments are started in the 40s of the XX century and are connected with the names of S. Rubinstein, B. Teplov, B. Ananyiev, O. Leontiev, V. Miasischev and others. The biggest spread and recognition was gained by the definition of abilities by B. Teplov as those of individually-psychological specificities which distinguish one person from the other. For contemporary state of research of the abilities problem the distinctive system approach is the one from the position of functional systems (V. Shadrikov). In the modern world the problem of abilities of people is gaining a new topicality and a new meaning in the context of meritocratic education, aimed at the early exposure and creating a special system of education for children with signs of academic, intellectual and social abilities and talent. The subject of the further research is justification of methodological approaches to the problem of improving the readiness of primary school teachers to work with gifted pupils.