Addressing Indigenous rights and interests in genetic resources has become increasingly challenging in an open science environment that promotes unrestricted access to genomic data. Although Indigenous experiences with genetic research have been shaped by a series of negative interactions, there is increasing recognition that equitable benefits can only be realized through greater participation of Indigenous communities. Issues of trust, accountability and equity underpin Indigenous critiques of genetic research and the sharing of genomic data. This Perspectives article highlights identified issues for Indigenous communities around the sharing of genomic data and suggests principles and actions that genomic researchers can adopt to recognize community rights and interests in data.