Prompted by developments in human genetics, a recurrent bioethical question concerns a person's 'right to know' and 'right not to know' about genetic information held that is intrinsically related to or linked to them. In this paper, we will revisit the claimed rights in relation to two particular test cases. One concerns the rights of the 500,000 participants in UK Biobank (UKB) whose biosamples, already having been genotyped, will now be exome sequenced, and the other concerns the rights of pregnant women (and their children) who undergo non-invasive prenatal testing (NIPT)-a simple blood test that can reveal genetic information about both a foetus and its mother. This two-part paper is in four principal sections. First, we sketch the relevant features of our two test cases. Secondly, we consider the significance of recent legal jurisprudence in the UK and Singapore. Thirdly, we consider how, the jurisprudence apart, the claimed rights might be grounded. Fourthly, we consider the limits on the rights. We conclude with some short remarks about the kind of genetically aware society that we might want to be and how far there is still an opportunity meaningfully to debate the claimed rights.