This present book collects a distinguished selection of contributions by scholars who participated as speakers or as visiting scientists in the intensive programme Puglia Summer Trimester 2023 took place in Bari, Italy, from April to July 2023, and also includes contributions by further scholars who are expert in related fields. The programme was structured around a series of main meetings, including a general conference and a summer school, supplemented by the local presence and activities of an amount of visiting scientists. Additionally, efforts were made to disseminate and popularise mathematics among schools and the general public, with the aim of extending the programme's impact beyond the immediate academic sphere. Each chapter, in the form of retrospective reviews, overviews on recent developments, announcements and comments of new results, as well as outlooks on future perspectives, represents some of the main scientific instances of the trimester in Bari. The trimester was actually focussed on a spectrum of mathematical problems, directly stemming or inspired from a variety of physical domains, involving singular modelling, asymptotic and emergent phenomena, singular interactions, non-trivial limit effects. Natural backgrounds are quantum physics, cold atom physics, soft matter physics, with methods and tools, suitably adapted to such singular settings, spanning across operator and spectral theory, functional analysis, probability, differential geometry, partial differential equations, and numerical analysis.