Background: Evidence for the transmission of non-genetic information from father to offspring is rapidly accumulating. While the impact of chemical and physical factors such as toxins or diet on the fitness of the parents and their offspring have been studied extensively, the importance of behavioural and social circumstances has only recently been recognised. Behavioural traits such as personality characteristics can be relatively stable, and partly comprise a genetic component but we know little about the non-genetic transmission of plastic behavioural traits from parents to offspring. We investigated the relative effect of personality and of social dominance as indicators at the opposite ends of the plasticity range on offspring behaviour in the zebrafish (Danio rerio). We assessed male boldness, a behavioural trait that has previously been shown previously to possess genetic underpinnings, and experimentally manipulated male social status to assess the association between the two types of behaviour and their correlation with offspring activity. Results: We found a clear interaction between the relatively stable and putative genetic effects based on inherited differences in personality and the experimentally induced epigenetic effects from changes in the social status of the father on offspring activity. Conclusions: Our study shows that offspring behaviour is determined by a combination of paternal personality traits and on-genetic effects derived from the social status of the father.