•Criminality is complex, and explanations that reduce offending behaviour to a genetic or neural level may be inappropriate and overly simplistic.•Whilst crime does appear to run in families, so do emotional instability, mental illness, social deprivation and poverty. This makes it difficult to disentangle the effects of genes and neural influences from other possible factors.•Additionally, whilst there is often a difference in concordance rates between MZ and DZ twins, MZ twins do not show 100% concordance.