Colloidal nanocrystals (NCs) have become an established optical material with applications in opto-electronics, nanophotonics and quantum technologies. NCs combine a suitability for solution-based processing and tunable materials properties, for example through the NC size or dimensionality (0D, 1D, 2D). Such characteristics have a direct impact on the excitons and excitonic complexes created in NCs by light absorption, and understanding the different exciton states, i.e., the exciton fine structure (EFS) is key for successful application development. Conex aims at utilizing the latest generation of NCs to extend the understanding of EFS towards new materials (III-V vs II-VI NCs) and explore the impact of dimensionality (2D vs. 0D), crystallinity (zinc blende vs. wurtzite) and the NC surface on exciton properties. The research will address both NC ensembles using linear and nonlinear optical spectroscopy and single NCs using micro-photoluminescence spectroscopy to map the charateristics of excitons and multiexciton complexes. In both cases, magneto-spectroscopy will be used to identify the various exciton states. Moreover, wide range of experimental data will be used to further develop the theory of excitons in nanoscale semiconductors to obtain internally consistent models that include exciton-surface interaction. CONEX results will help researchers develop classical and quantum light emitters, solar cells, displays, biological markers and sensors based on colloidal NCs.