RNA and DNA molecules can form complex, three-dimensional folded structures that have surprisingly sophisticated functions, including catalysing chemical reactions and controlling gene expression. Although natural nucleic acids make occasional use of these advanced functions, the true potential for sophisticated function by these biological polymers is far greater. An important challenge for biochemists is to take RNA and DNA beyond their proven use as polymers that form double-helical structures. Molecular engineers are beginning to harness the power of nucleic acids that form more complex three-dimensional structures, and apply them as tools for exploring biological systems and as therapeutics.