It is often computationally expensive to monitor structural health using computer models. This time-consuming process can be relieved using surrogate models, which provide cheap-to-evaluate metamodels to replace the original expensive models. Because of their high accuracy, simplicity, and efficiency, Artificial Neural Networks (ANNs) have gained considerable attention in this area. This paper reviews the application of ANNs as surrogates for structural health monitoring in the literature. Moreover, the review contains fundamental information, detailed discussions, wide comparisons, and suggestions for future research. Surrogates in this literature review are divided into parametric and nonparametric models. In the past, nonparametric models dominated this field, but parametric models have gained popularity in the recent decade. A parametric surrogate is commonly supplied with metaheuristic algorithms, and can provide high levels of identification. Recurrent networks, instead of traditional ANNs, have also become increasingly popular for nonparametric surrogates.