In recent years, there has been a growing interest in synthetic data for several computer vision applications, such as automotive, detection and tracking, surveillance, medical image analysis and robotics. Early use of synthetic data was aimed at performing controlled experiments under the analysis by synthesis approach. Currently, synthetic data are mainly used for training computer vision models, especially deep learning ones, to address well-known issues of real data, such as manual annotation effort, data imbalance and bias, and privacy-related restrictions. In this work, we survey the use of synthetic training data focusing on applications related to video surveillance, whose relevance has rapidly increased in the past few years due to their connection to security: crowd counting, object and pedestrian detection and tracking, behaviour analysis, person re-identification and face recognition. Synthetic training data are even more interesting in this kind of application, to address further, specific issues arising, e.g., from typically unconstrained image or video acquisition conditions and cross-scene application scenarios. We categorise and discuss the existing methods for creating synthetic data, analyse the synthetic data sets proposed in the literature for each of the considered applications, and provide an overview of their effectiveness as training data. We finally discuss whether and to what extent the existing synthetic data sets mitigate the issues of real data, highlight existing open issues, and suggest future research directions in this field.