The article analyzes the foreign publications for the last five years on the research of anxiety, its different types and their relationship with cognitive abilities, academic achievements, emotional and social development of schoolchildren. The main attention is paid to studies of the anxiety in intellectually gifted children as well as the contradictions in ideas about giftedness as a special vulnerability to problems or as a resource for their successful overcoming. These contradictions emerge mainly due to differences in the definitions of giftedness and gifted children. It is shown that intellectually gifted children face the same age-related development problems as their peers, and cope with these problems no less, and sometimes even more successfully. At the same time, the recognition of unique problems for intellectually gifted children that are associated with increased anxiety and other emotional disorders, as well as the need for psychological support, is emphasized.