DNA computing and DNA data storage are emerging fields that are unlocking new possibilities in information technology and diagnostics. These approaches use DNA molecules as a computing substrate or a storage medium, offering nanoscale compactness and operation in unconventional media (including aqueous solutions, water-in-oil microemulsions and self-assembled membranized compartments) for applications beyond traditional silicon-based computing systems. To build a functional DNA computer that can process and store molecular information necessitates the continued development of strategies for computing and data storage, as well as bridging the gap between these fields. In this Review, we explore how DNA can be leveraged in the context of DNA computing with a focus on neural networks and compartmentalized DNA circuits. We also discuss emerging approaches to the storage of data in DNA and associated topics such as the writing, reading, retrieval and post-synthesis editing of DNA-encoded data. Finally, we provide insights into how DNA computing can be integrated with DNA data storage and explore the use of DNA for near-memory computing for future information technology and health analysis applications.