In recent times, the reduced cost of DNA sequencing has resulted in a plethora of genomic data that is being used to advance biomedical research and improve clinical procedures and healthcare delivery. These advances are revolutionizing areas in genome-wide association studies (GWASs), diagnostic testing, personalized medicine and drug discovery. This, however, comes with security and privacy challenges as the human genome is sensitive in nature and uniquely identifies an individual. In this article, we discuss the genome privacy problem and review relevant privacy attacks, classified into identity tracing, attribute disclosure and completion attacks, which have been used to breach the privacy of an individual. We then classify state-of-the-art genomic privacy-preserving solutions based on their application and computational domains (genomic aggregation, GWASs and statistical analysis, sequence comparison and genetic testing) that have been proposed to mitigate these attacks and compare them in terms of their underlining cryptographic primitives, security goals and complexities—computation and transmission overheads. Finally, we identify and discuss the open issues, research challenges and future directions in the field of genomic privacy. We believe this article will provide researchers with the current trends and insights on the importance and challenges of privacy and security issues in the area of genomics.