In the last two decades, network science has revolutionized our understanding of diverse biological, social, and technological systems, becoming a cornerstone across various scientific and technological domains. Traditional network models, however, predominantly focus on pairwise interactions, often overlooking the complexity of higher-order interactions (HOIs) that involve more than two elements. This oversight is significant, as emerging evidence increasingly demonstrates that HOIs are not only prevalent but also critically influence the emergent behaviors of complex systems, thereby impacting our ability to comprehend and forecast their dynamics. RUNES aims to pioneer a novel paradigm in network science, tailored to the intricacies of higher-order systems. This project will develop a holistic mathematical, methodological, and empirical framework to effectively understand, predict, and manipulate HOIs. This will involve redefining existing network models to incorporate higher-order interactions, combining information-theoretic and topological perspectives, and formulating new dynamical processes on these advanced substrates. Concurrently, RUNES will innovate in developing model-informed statistical methods for inferring HOIs from data, enabling us to discern their unique structures and functions. This framework will be applied in two groundbreaking contexts: firstly, in experimental behavioral science, where we will quantify the impact of HOIs in controlled human group experiments; secondly, in neuroscience, where we will investigate the higher-order functional topology of the brain across various conditions, ages, and species. RUNES’ new science of higher-order real-world complex systems will directly impact how we think about and deal with structural and dynamical patterns in many contexts, including epidemiology, artificial intelligence, economy and medicine.