Within the myriad articles about participants' opinions of genomics research, the views of a distinct group - people with a rare disease (RD) - are unknown. It is important to understand if their opinions differ from the general public by dint of having a rare disease and vulnerabilities inherent in this. Here we document RD patients' attitudes to participation in genomics research, particularly around large-scale, international data and biosample sharing. This work is unique in exploring the views of people with a range of rare disorders from many different countries. The authors work within an international, multidisciplinary consortium, RD-Connect, which has developed an integrated platform connecting databases, registries, biobanks and clinical bioinformatics for RD research. Focus groups were conducted with 52 RD patients from 16 countries. Using a scenario-based approach, participants were encouraged to raise topics relevant to their own experiences, rather than these being determined by the researcher. Issues include wide data sharing, and consent for new uses of historic samples and for children. Focus group members are positively disposed towards research and towards allowing data and biosamples to be shared internationally. Expressions of trust and attitudes to risk are often affected by the nature of the RD which they have experience of, as well as regulatory and cultural practices in their home country. Participants are concerned about data security and misuse. There is an acute recognition of the vulnerability inherent in having a RD and the possibility that open knowledge of this could lead to discrimination.