Complex networks have become the main paradigm for modelling the dynamics of complex interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units.

Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the multilayered real organization of many social, biological and man-made systems. At the same time, higher-order observables, typically topological or information-theoretic in nature and often sharing the same simplicial language, have been gathering  attention for their capacity to capture properties of complex systems that are invisible to standard statistical descriptions. This had led to a certain confusion between these two facets, mechanisms on one side, phenomena on the other.

Here, using recent examples from both dynamical models and neuroimaging analysis, I highlight collective behaviours induced by higher-order interactions, the difficulty in linking data and models through recent advances in topological data analysis and higher-order information theory, and finally outline key open questions for the physics of higher-order complex systems.