Kuwait has a history over a number of decades of identifying the most academically able school students and, in recent years, this has led to the establishment of an enrichment programme for those students described as ‘gifted’. The process of selection is basically according to academic achievement and the enrichment provision aims to give them special activities for high thinking skills through a specially designed syllabus. This study seeks to explore the cognitive characteristics of such high achievement students in middle school (ages 13-15) as well as a wider range of students, the work being carried out in the State of Kuwait. The study discusses the nature of giftedness and how it might be defined, moving on to look at ways by which selection can be considered. The study was conducted on a total sample of 2169 students, from middle schools in State of Kuwait. The research was carried out in three experiments. In the first experiment, the relationship between cognitive characteristics (working memory capacity, field dependency, divergency, and visual-spatial characteristics) and performance in six subjects are explored with a large sample containing a high proportion of very able students aged about 13. The second experiment investigated the relationship between cognitive characteristics and self awareness, along with school performance with two samples: the first one selected students from the first experiment who scored highly in most of the cognitive characteristics (124 from grade 8); the second group was 299 students in grade 7 from the same 15 Kuwaiti middle schools who participated in experiment one. In the third experiment, samples in this experiment were drawn from grade 7, grade 8 and grade 9, the aim of this experiment being to examine the relationship between cognitive characteristics (divergency, convergency; and visual-spatial disabilities), using freshly designed tests for visual-spatial abilities and convergency.