This special issue of the European journal of applied mathematics features research articles that involve the application of mathematical methodologies to the modelling of a broad range of problems related to crime and security. Some specific topics in this issue include recent developments in mathematical models of residential burglary, a dynamical model for the spatial spread of riots initiated by some triggering event, the analysis and development of game-theoretic models of crime and conflict, the study of statistically based models of insurgent activity and terrorism using real-world data sets, models for the optimal strategy of police deployment under realistic constraints, and a model of cyber crime as related to the study of spiking behaviour in social network cyberspace communications. Overall, the mathematical and computational methodologies employed in these studies are as diverse as the specific applications themselves and the scales (spatial or otherwise) to which they are applied. These methodologies range from statistical and stochastic methods based on maximum likelihood methods, Bayesian equilibria, regression analysis, self-excited Hawkes point processes, agent-based random walk models on networks, to more traditional applied mathematical methods such as dynamical systems and stability theory, the theory of Nash equilibria, rigorous methods in partial differential equations and travelling wave theory, and asymptotic methods that exploit disparate space and time scales.