Recent progress in genomics is bringing genetic testing to the masses. Participatory public initiatives are underway to sequence the genome of millions of volunteers, and a new market is booming with a number of companies like 23andMe and AncestryDNA offering affordable tests directly to consumers. Consequently, news, experiences, and views on genetic testing are increasingly shared and discussed online and on social networks like Twitter. In this paper, we present a large-scale analysis of Twitter discourse on genetic testing. We collect 302K tweets from 113K users, posted over 2.5 years, by using thirteen keywords related to genetic testing companies and public initiatives as search keywords. We study both the tweets and the users posting them along several axes, aiming to understand who tweets about genetic testing, what they talk about, and how they use Twitter for that. Among other things, we find that tweets about genetic testing originate from accounts that overall appear to be interested in digital health and technology. Also, marketing efforts as well as announcements, such as the FDA's suspension of 23andMe's health reports, influence the type and the nature of user engagement.Finally, we report on users who share screenshots of their results, and raise a few ethical and societal questions as we find evidence of groups associating genetic testing to racist ideologies.