Ants form colonies, birds flock, mobile networks synchronize, and a consensus may emerge from the interaction of diverse human opinions. These are simple examples of collective dynamics in which small-scale interactions lead to the emergence of larger-scale patterns.

We will survey recent mathematical developments in collective dynamics. Three levels of descriptions are considered: agent-based dynamics on graphs, Vlasov-type kinetic description, and large-crowd hydrodynamics. The dynamics is governed by different protocols of pairwise interactions. Different models for such protocols of attraction, alignment, and repulsion go back to the influential works of Reynolds, Krause, Vicsek, and Cucker & Smale.

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