DNA computing is an emerging field that aims at enabling more efficient data storage and processing. One principle of DNA computing is to encode some information (e.g., image, video, programming scripts) into a digital DNA-like sequence and then synthesize the corresponding DNA molecule. Synthesizing this molecule using digital or real human genomic fragments theoretically opens the possibility for privacy attacks, which have been demonstrated on a large array of human genomic data. These privacy attacks aim at breaching the privacy of DNA samples, allowing an attacker to discover privacy-critical information from the partial or complete DNA information of an individual. In the context of DNA computing, novel privacy attacks will certainly emerge and could consist in discovering a part of a particular script or video that is privacy-critical. It is therefore important to consider whether privacy attacks and defense mechanisms can be used when manipulating genomic data. First, this chapter provides the background about genomic data, and its modern generation and processing. It then provides a survey on known genomic privacy attacks, and presents the privacy-enhancing technologies that have been designed to protect genomic data. Later, this chapter also introduces the current trust management methods one can rely on to further secure DNA storage and processing methods, before discussing how DNA computing currently relates to those attacks and privacy-preserving technologies. Finally, this chapter presents future research avenues.