Today the world is becoming increasingly “sensorized”, from mobile phones and personal health monitors to routine medical imaging and satellite surveillance. There is an exponential growth in the generation and consumption of data and an ever increasing demand for faster and yet more sophisticated sensing and imaging systems. The need to reconcile the growing demands made of modern sensing and data systems with the fundamental resource limitations, both in terms of sensor acquisition and computation, provides new fundamental mathematical and computational challenges. These challenges belong to the realms of signal processing and information theory which are concerned with the conversion of measurements to information. The proposed research will push the boundaries on what can be inferred from sensors and data, developing and extending the emerging field of compressed sensing theory. We will also go beyond this and explore the trade-off between computation and sensing, challenging the notion that better sensing and imaging can only come at a high computational cost – research that will also be valuable for the development of scalable processing solutions for an array of challenges in data science. In our work at the University of Edinburgh we are already exploiting this theory and its extensions to develop new advanced medical imaging techniques for Magnetic Resonance Imaging (MRI) and X-ray computer tomography (CT), resulting in better imaging performance with lower doses and in faster scan times. In the defence domain we are using compressed sensing to devise new algorithms for radar imaging and chemical sensing in association with the UK defence science and technology laboratories (Dstl), offering better assessment of threats e.g. identifying covert movement of weapons or the detection of improvised explosives.