DNA outperforms most conventional storage media in terms of information retention time, physical density, and volumetric coding capacity. Advances in synthesis and sequencing technologies have enabled implementations of large synthetic DNA databases with impressive storage capacity and reliable data recovery. Several robust DNA storage architectures featuring random access, error correction, and content rewritability have been constructed with the potential for scalability and cost reduction. We survey these recent achievements and discuss alternative routes for overcoming the hurdles of engineering practical DNA storage systems. We also review recent exciting work on in vivo DNA memory including intracellular recorders constructed by programmable genome editing tools. Besides information storage, DNA could serve as a versatile molecular computing substrate. We highlight several state-of-the-art DNA computing techniques such as strand displacement, localized hybridization chain reactions, and enzymatic reaction networks. We summarize how these simple primitives have facilitated rational designs and implementations of in vitro DNA reaction networks that emulate digital/analog circuits, artificial neural networks, or nonlinear dynamic systems. We envision these modular primitives could be strategically adapted for sophisticated database operations and massively parallel computations on DNA databases. We also highlight in vivo DNA computing modules such as CRISPR logic gates for building scalable genetic circuits in living cells. To conclude, we discuss various implications and challenges of DNA-based storage and computing, and we particularly encourage innovative work on bridging these two areas of research to further explore molecular parallelism and near-data processing. Such integrated molecular systems could lead to far-reaching applications in biocomputing, security, and medicine.