In the late 60s a new area emerged in mathematical physics known as "Integrable Systems". Ideas and techniques of "Integrability" have had a significant impact in several areas of mathematics, science and engineering, from the proof of the Schottky problem in algebraic geometry, to optical communications. In this lecture, two such implications will be reviewed: (a) A novel method for analysing boundary value problems, which unifies the fundamental contributions to the analytical solution of PDEs of Fourier, Cauchy and Green, and also constructs a non-linearization of some of these results. This method has led to the emergence of new numerical techniques for solving linear elliptic PDEs in polygonal domains. (b) A new approach for solving the inverse problems arising in certain important medical imaging techniques, including Single Photon Emission Computerised Tomography (SPECT).