As detailed in its flagship report, Genome UK, the UK government recognises the vital role that broad public engagement across whole populations plays in the field of genomics. However, there is limited evidence about how to do this at scale. Most public audiences do not feel actively connected to science, are oftenunsure of the relevance to their lives and rarely talk to their family and friends about; we term this dis-connection a 'disengaged public audience'. We use a narrative review to explore: (i) UK attitudes towards genetics and genomics and what may influence reluctance to engage with these topics; (ii) innovative public engagement approaches that have been used to bring diverse public audiences into conversations about the technology. Whilst we have found some novel engagement methods that have used participatory arts, film, social media and deliberative methods, there is no clear agreement on best practice. We did not find a consistently used, evidence-based strategy for delivering public engagement about genomics across diverse and broad populations, nor a specific method that is known to encourage engagement from groups that have historically felt (in terms of perception) and been (in reality) excluded from genomic research. We argue there is a need for well-defined, tailor-made engagement strategies that clearly articulate the audience, the purpose and the proposed impact of the engagement intervention. This needs to be coupled with robust evaluation frameworks to build the evidence-base for population-level engagement strategies.