The problem of school bullying is usually identified with the bullies, in a way that bullies themselves are considered to be "the problem of the school". But are bullies a problem or a challenge for schools to change their philosophy and practice? This study is an attempt to examine whether bullying could be considered as a form of aggressive behaviour, to investigate the situation in State Primary Schools in Cyprus and to explore the developmental history of nine 11-year-old bullies in terms of their psychological and social characteristics. The study employed both qualitative and quantitative research methods and provided support for the view that bullying is a form of aggressive behaviour as all the factors that are reported in the relevant literature about aggression seem to play a role in its development, albeit to a different extent and in different combinations. Many factors related to the bullies’ psychological and social characteristics are involved in an act of bullying and consequently each has to be seriously taken into consideration, both individually and in combination, in any effort aiming at the prevention or inhibition of the problem in schools.