In recent years, Whole Genome Sequencing (WGS) evolved from a futuristic-sounding research project to an increasingly affordable technology for determining complete genome sequences of complex organisms, including humans. This prompts a wide range of revolutionary applications, as WGS promises to improve modern healthcare and provide a better understanding of the human genome -- in particular, its relation to diseases and response to treatments. However, this progress raises worrisome privacy and ethical issues, since, besides uniquely identifying its owner, the genome contains a treasure trove of highly personal and sensitive information. In this article, after summarizing recent advances in genomics, we discuss some important privacy issues associated with human genomic information and identify a number of particularly relevant research challenges.