Developing 1 new science produces technological benefits: this is so often repeated it is a commonplace nowadays. In our opinion the converse statement, i.e. that cuttingedge technology provides substantial stimuli for scientific innovation, has not to be underestimated either. Hence, the interaction between science and technology has not to be regarded as unilateral/one-way. In fact (as summarized in Fig. 1.1), the investigation of theoretical models describing some known phenomena can lead to the development of new scientifically designed devices which, in turn, might unveil some not-yetdiscovered phenomena. This process of scientific modeling, designing and experimental discoveries could in principle keep going indefinitely. Of note, the process leads to the discovery of progressively "higher order" phenomena, which become accessible only after some necessary "lower order" modeling and discoveries. Apparent examples of very high-order phenomena are easily found in current research (to mention a recent and widely known case) in quantum gravity, concerning the discovery of gravitational waves.It is a matter of fact that emerging technologies and development of the exact sciences have a close relation. Actually, it is a leitmotiv in the History of Science that new technological possibilities lead to new phenomenological evidences, putting in crisis any existing paradigm and gradually leading to a totally new one. The birth of scientific technology in the Hellenistic World, the rise of modern mechanics in the age of Galileo, and the development of thermodynamics in the early nineteenth century are relevant examples of this phenomenon. In all these cases, it is by now well-established among historians of science that a significant conceptual revolution occurred driven by technological reasons. The main goal of these successful new ideas, that nowadays after a long and troublesome process we call classical physics, was to design and describe new technology (as for example bombards, steam engines, or catapults). Let us examine in more detail one of the above-mentioned examples: the revolution in the conception of mechanics due to the results of Galileo and his school. Simplifying necessarily a complex matter, we can say that, while within the Peripatetic school (the philosophic tradition based on Aristotle) the motion of objects like the projectiles of bombards does not obey the same laws that govern celestial mechanics. Galileo managed to include phenomena