Biological systems illustrate how complex and dynamic physical and chemical interactions between many different components can produce organized structures across length scales, ranging from angstroms to hundreds of meters, and precise temporal control over diverse material dynamics. While mechanisms for pattern formation such as reaction-diffusion processes, message passing, or rule-based assembly have been studied extensively using mathematical models, it can be difficult to create synthetic materials that implement these mechanisms. Here, we describe how DNA nanotechnology techniques make it possible to systematically build systems capable of complex self-organization or pattern formation across scales. DNA-programmed short-range interactions can be used to build aperiodic crystals and assemblies with long-range order, form patterns using reaction-diffusion and chemical message passing, and create self-organizing or stimulus-responsive amorphous materials, including gels or cell-sized compartments. Exploiting principles from self-organization using DNA-based interactions makes it possible to build materials with complex long-range order and intelligent spatiotemporal responses to a variety of stimuli using relatively simple bottom-up methods.