Early forecasts of dengue are an important tool for disease mitigation. Neural networks are powerful predictive models that have made contributions to many areas of public health. In this study, we reviewed the application of neural networks in the dengue forecasting literature, with the objective of informing model design for future work. Following PRISMA guidelines, we conducted a systematic search of studies that use neural networks to forecast dengue in human populations. We summarized the relative performance of neural networks and comparator models, architectures and hyper-parameters, choices of input features, geographic spread, and model transparency. Sixty two papers were included. Most studies implemented shallow feed-forward neural networks, using historical dengue incidence and climate variables. Prediction horizons varied greatly, as did the model selection and evaluation approach. Building on the strengths of neural networks, most studies used granular observations at the city level, or on its subdivisions, while also commonly employing weekly data. Performance of neural networks relative to comparators, such as tree-based supervised models, varied across study contexts, and we found that 63% of all studies do include at least one such model as a baseline, and in those cases about half of the studies report neural networks as the best performing model. The studies suggest that neural networks can provide competitive forecasts for dengue, and can reliably be included in the set of candidate models for future dengue prediction efforts. The use of deep networks is relatively unexplored but offers promising avenues for further research, as does the use of a broader set of input features and prediction in light of structural changes in the data generation mechanism.