Rapid advances in human genomics are enabling researchers to gain a better understanding of the role of the genome in our health and well-being, stimulating hope for more effective and cost efficient healthcare. However, this also prompts a number of security and privacy concerns stemming from the distinctive characteristics of genomic data. To address them, a new research community has emerged and produced a large number of publications and initiatives. In this paper, we rely on a structured methodology to contextualize and provide a critical analysis of the current knowledge on privacy-enhancing technologies used for testing, storing, and sharing genomic data, using a representative sample of the work published in the past decade. We identify and discuss limitations, technical challenges, and issues faced by the community, focusing in particular on those that are inherently tied to the nature of the problem and are harder for the community alone to address. Finally, we report on the importance and difficulty of the identified challenges based on an online survey of genome data privacy experts.