Perfectly aligned with the vision of the EuroHPC Joint Undertaking, the SparCity project aims at creating a supercomputing framework that will provide efficient algorithms and coherent tools specifically designed for maximising the performance and energy efficiency of sparse computations on emerging HPC systems, while also opening up new usage areas for sparse computations in data analytics and deep learning. The framework enables comprehensive application characterization and modeling, performing synergistic node-level and system-level software optimizations. By creating a digital SuperTwin, the framework is also capable of evaluating existing hardware components and addressing what-if scenarios on emerging architectures and systems in a co-design perspective. To demonstrate the effectiveness, societal impact, and usability of the framework, the SparCity project will enhance the computing scale and energy efficiency of four challenging real-life applications that come from drastically different domains, namely, computational cardiology, social networks, bioinformatics and autonomous driving. By targeting this collection of challenging applications, SparCity will develop world-class, extreme scale and energy-efficient HPC technologies, and contribute to building a sustainable exascale ecosystem and increasing Europe’s competitiveness.