Contemporary research in mathematical logic shows increasing interactions between Model Theory (MT), Set Theory (ST), and Computability Theory (CT), guided by inner developments which progressively found applications to larger and larger areas of mathematics. Problems originating from MT lead naturally to ST and CT questions, while the forcing method, originally developed within ST, and the tecniques of descriptive set theory find applications in MT and CT. Our project is inserted in this general setting. Its common aim is to classify mathematical structures and theories, study their definable sets and relations, and gauge their complexity according to various hierarchies. We plan to tackle a number of key questions in MT, ST, CT with implications also in other domains of the mathematical sciences. New developments have seen the important contribution of many of the participants of the project, and surprising connections between different lines of research have already emerged.