In recent years dual-unitary circuits have emerged as minimal models for chaotic quantum many-body systems in which the dynamics of correlations and entanglement remains tractable. The building blocks of these circuits are gates that are unitary in both time and space. After an introduction to these circuits I will extend the notion of dual-unitarity to biunitarity, which allows for a richer variety of building blocks and circuit dynamics, as well as unifying different notions of ‘dual-unitarity’. The resulting interactions are governed by quantum combinatorial data, which I will precisely characterize using a graphical 2-categorical framework based on the ‘shaded calculus’. These generalized circuits remain exactly solvable and we show that they retain the attractive features of the original dual-unitary models, with exact results for correlations functions, maximal entanglement growth and exact thermalization.