This conference is intended as a follow-up of other highly successful events held in Europe (ESF Exploratory Workshop, Prague 2010; ICMS Workshop, Edinburgh 2013; AIM Workshop, San Jose 2015). While the above workshops were deliberately smaller, and less formal meetings focused on open problems in the field, this event at CIRM will be a full-scale conference to which previous activities naturally result. The main goal is to facilitate interdisciplinary collaborations across the mathematical analysis and mathematical physics community, with a central role being played by the theory of non-self-adjoint operators. The meeting will address state-of-the-art techniques for the mathematically rigorous analysis of non-self-adjoint phenomena encountered in main stream and newly developing fields of physics. An emphasis will be put on non-self-adjoint operators related to problems in superconductivity, graphene, repeated interaction quantum systems, hydrodynamics, quantum mechanics with non-Hermitian operators, optics and metamaterials. Experts in operator theory, microlocal analysis, and mathematical physics will be brought together to address challenging open problems involving the rigorous mathematical framework of the physical theories.