Final Report Summary - MODULI (Indo European Collaboration on Moduli Spaces) Moduli spaces are basic and central objects that emerge in the most natural classification problems in geometry. Their importance has been emphasized over the years due to the relation of these spaces with areas of mathematics so diverse as algebraic geometry, differential geometry, topology, arithmetic geometry, algebra, global analysis and, perhaps more surprisingly, with theoretical physics.The main objective of MODULI project was to establish new collaborations and reinforce existing collaborations between Europe and India in the field of moduli spaces, as well as training young researches. This was done through the exchange of both experienced and early stage researchers, and the organization of conferences, specialized workshops and schools.The project was a Marie Curie International Research Staff Exchange Scheme (IRSES) project in the 7th European Framework Program, with project nº 612534, and it run from 1 January 2014 to 31 December 2017. There were four European partners involved: Consejo Superior de Investigaciones Científicas-CSIC (Madrid, Spain), University of Oxford (Oxford, UK), Université Pierre et Marie Curie (Paris, France) and Aarhus Universitet (Aarhus, Denmark); and four Indian partners: Tata Institute for Fundamental Research (Mumbai, India), Chennai Mathematical Institute (Chennai, India), Institute of Mathematical Sciences (Chennai, India) and Indian Institute of Science (Bangalore, India). The project was structured around the following areas of this field:• moduli spaces and geometric structures.• moduli spaces and physics.• moduli spaces and arithmetic geometry.In all these areas, the project was extremely successful in obtaining new important results that have appeared in top international journals, as well as in strengthening existing collaborations and creating new ones. There has been also a big impact regarding the training and mobility of young researchers. The project has established, no doubt, a new basis of collaboration between Indian and European mathematicians working in the field.