Automated essay scoring (AES) is the task of automatically assigning scores to essays as an alternative to grading by humans. Although traditional AES models typically rely on manually designed features, deep neural network (DNN)-based AES models that obviate the need for feature engineering have recently attracted increased attention. Various DNN-AES models with different characteristics have been proposed over the past few years. To our knowledge, however, no study has provided a comprehensive review of DNN-AES models while introducing each model in detail. Therefore, this review presents a comprehensive survey of DNN-AES models, describing the main idea and detailed architecture of each model. We classify the AES task into four types and introduce existing DNN-AES models according to this classification.