Final Report Summary - FIRST (Fronts and Interfaces in Science and Technology) Within FIRST (http://www.mat.ucm.es/~FIRST/ ) 41 ESR contracts, distributed among 24 young researchers, have been dedicated to the study of a large spectrum of problems in nonlinear partial differential equations related to many models involving fronts and interfaces in science and technology. Space-time phenomena giving rise to fronts and interfaces are central issues in many contexts of Science (understood here in the largest possible meaning) and Technology (including aspects related with engineering and industry). A selection of some of the most challenging ones, very often of multi-disciplinary character and new, was given here.An important feature of the project was a close contact between the experimental researches and the corresponding theoretical analysis and interpretation. For instance, Non Linear Partial Differential Equations (NL-PDEs) emerged recently as a major and basic tool being useful in a wide variety of Image Processing tasks. As one of the main goals of this proposal we plan to bring together engineering researchers whose work led to the development of numerical algorithms to solve complex systems of NL-PDEs, with mathematicians who have the expertise to address questions of well-posedness uniqueness of solutions and stability of solutions with respect to coefficient perturbations and small changes in the initial/boundary conditions. Obviously, careful mathematical modelling of the physical processes encountered in many other applications often leads to intricate systems formulated in terms of NL-PDEs. These systems sometimes involve stochastic terms and/or non-local terms representing delay, hysteresis or long-range-interaction effects.A selection of some of the most challenging , space-time phenomena giving rise to fronts and interfaces most often of multi-disciplinary character, were pursued by a team of 10 Full Partner Members (8 universities of different countries and two industries, Egis Environnement (France) and Siemens (Germany)) with the help of 4 Associate Partners. The project was divided into three general Work Packages A: Image Processing, B: Patterns in Complex Reaction-Diffusion Systems and C: Interfaces, Control and Inverse Methods in Technology Problems. The training was proposed with an interdisciplinary orientation. Besides the local training courses organized by FIRST, to complement the network's capacity to transfer new knowledge and strengthen supervision of the network-wide training activities, six seniors Visiting Scientists were recruited for periods of one month. During the project 13 thematic workshops were organized, one by country. There was three major Conferences (Starting Conference (France), Mid Term Conference (United Kingdom) and Final Network Conference (Spain). The workshops allowed to establish major research linkages between the local teams and the other teams in the network. Two workshops, organized by Egis and Siemens, were devoted to complementary skills. Another one was organized by the ESRs (Eindhoven 2013).The results oobtained by the ESRs have been presented at many other different conferences, and published (or under publication). The complete list of publication can be found on http://www.mat.ucm.es/~FIRST/ . The large spectrum of properties, techniques and methods, was a fundamental part of their training and will play an important role in their future career.Apart from the laboratory training, the fellows have received extra training during the semestrial seminars where scientific staff, visiting scientists and external researchers have presented talks and tutorials. They have followed some Complementary Skills workshops with two industrial partners, Siemens and EGIS and for research management in Eindhoven. Most of the fellows are now defending or preparing the defense of their PhD. Among them, some have already been hired as scientific staff. Most of them are now interested in pursuing their career as post-doctoral fellows, and hope to later apply for academic positions. The benefit to the European scientific community was quite significant: our goal training different ESRs by FIRST was not only to produce researchers with broad backgrounds qualified for academic careers but also with an understanding of science and with an excellent perspective for a career outside academia.