The task of information processing, or computation, can be performed by natural and man-made 'devices'. Man-made computers are made from silicon chips, whereas natural 'computers', such as the brain, use cells and molecules. Computation also occurs on a much smaller scale in regulatory and signalling pathways in individual cells and even within single biomolecules. Indeed, much of what we recognize as life results from the remarkable capacity of biological building blocks to compute in highly sophisticated ways. Rational design and engineering of biological computing systems can greatly enhance our ability to study and to control biological systems. Potential applications include tissue engineering and regeneration and medical treatments. This Review introduces key concepts and discusses recent progress that has been made in biomolecular computing.