Animal groups are said to make consensus decisions when group members come to agree on the same option. Consensus decisions are taxonomically widespread and potentially offer three key benefits: maintenance of group cohesion, enhancement of decision accuracy compared with lone individuals and improvement in decision speed. In the absence of centralized control, arriving at a consensus depends on local interactions in which each individual's likelihood of choosing an option increases with the number of others already committed to that option. The resulting positive feedback can effectively direct most or all group members to the best available choice. In this paper, we examine the functional form of the individual response to others' behaviour that lies at the heart of this process. We review recent theoretical and empirical work on consensus decisions, and we develop a simple mathematical model to show the central importance to speedy and accurate decisions of quorum responses, in which an animal's probability of exhibiting a behaviour is a sharply nonlinear function of the number of other individuals already performing this behaviour. We argue that systems relying on such quorum rules can achieve cohesive choice of the best option while also permitting adaptive tuning of the trade-off between decision speed and accuracy.