This paper examines some arguments which deny the existence of an individual right to remain ignorant about genetic information relating to oneself--often referred to as 'a right to genetic ignorance' or, more generically, as 'a right not to know'. Such arguments fall broadly into two categories: 1) those which accept that individuals have a right to remain ignorant in self-regarding matters, but deny that this right can be extended to genetic ignorance, since such ignorance may be harmful to others, particularly those to whom one is genetically related (the 'harm to others objection') and 2) those which contend that, even if genetic ignorance is only self harming, it is not something to which individuals can rationally or morally claim to have a 'right' at all, since they defend their claims on autonomy-respecting grounds and ignorance is inimical to autonomy (the 'incoherence objection'). I argue that defenders of a right not to know have some plausible responses to the 'harm to others objection', they and their opponents reach an impasse in which both sides are left voicing concerns about the paternalistic implications of the other's point of view. I conclude that defenders of a right not to know would, therefore, advance their position further by analysing it in terms of values other than those of autonomy and rights.