We give a general recipe for derandomising PAC-Bayesian bounds using margins, with the critical ingredient being that our randomised predictions concentrate around some value. The tools we develop traightforwardly lead to margin bounds for various classifiers, including linear prediction—a class that includes boosting and the support vector machine—single-hidden-layer neural networks with an unusual erf activation function, and deep ReLU networks. Further, we extend to partially-derandomised predictors where only some of the randomness is removed, letting us extend bounds to cases where the concentration properties of our predictors are otherwise poor.