This volume contains eight survey articles by the invited speakers of the 31st British Combinatorial Conference, held at Cardiff University in July 2026. Each article provides an overview of recent developments in a current hot research topic in combinatorics. Topics covered include random planar graphs, temporal graphs, domino tilings, extremal poset theory, asymptotic enumeration, graph homomorphisms, combinatorial rigidity theory, logic and model theory, matroids, and graph bootstrap percolation. The authors are among the world's foremost researchers on their respective topics, but their surveys are accessible to nonspecialist readers: they are written clearly, with little prior knowledge assumed, and with pointers to the wider literature. Taken together, these surveys give a snapshot of the research frontier in contemporary combinatorics, helping researchers and graduate students in mathematics and theoretical computer science to keep abreast of the latest developments in the field.

This volume contains eight survey articles by the invited speakers of the 31st British Combinatorial Conference, held at Cardiff University in July 2026. Each article provides an overview of recent developments in a current hot research topic in combinatorics. Topics covered include random planar graphs, temporal graphs, domino tilings, extremal poset theory, asymptotic enumeration, graph homomorphisms, combinatorial rigidity theory, logic and model theory, matroids, and graph bootstrap percolation. The authors are among the world's foremost researchers on their respective topics, but their surveys are accessible to nonspecialist readers: they are written clearly, with little prior knowledge assumed, and with pointers to the wider literature. Taken together, these surveys give a snapshot of the research frontier in contemporary combinatorics, helping researchers and graduate students in mathematics and theoretical computer science to keep abreast of the latest developments in the field.