Bullying comes in different forms, yet most previous genetically-sensitive studies have not distinguished between them. Given the serious consequences and the high prevalence of bullying, it is remarkable that the aetiology of bullying and its different forms has been under-researched. We present the first study to investigate the genetic architecture of bullying perpetration, bullying victimization, and their co-occurrence for verbal, physical and relational bullying. Primary-school teachers rated 8215 twin children on bullying perpetration and bullying victimization. For each form of bullying, we investigated, through genetic structural equation modelling, the genetic and environmental influences on being a bully, a victim or both. 34% of the children were involved as bully, victim, or both. The correlation between being a bully and being a victim varied from 0.59 (relational) to 0.85 (physical). Heritability was ~ 70% for perpetration and ~ 65% for victimization, similar in girls and boys, yet both were somewhat lower for the relational form. Shared environmental influences were modest and more pronounced among girls. The correlation between being a bully and being a victim was explained mostly by genetic factors for verbal (~ 71%) and especially physical (~ 77%) and mostly by environmental factors for relational perpetration and victimization (~ 60%). Genes play a large role in explaining which children are at high risk of being a victim, bully, or both. For victimization this suggests an evocative gene-environment correlation: some children are at risk of being exposed to bullying, partly due to genetically influenced traits. So, genetic influences make some children more vulnerable to become a bully, victim or both.