In the last few years, due to the continuous advancement of technology, human behavior detection and recognition have become important scientific research in the field of computer vision (CV). However, one of the most challenging problems in CV is anomaly detection (AD) because of the complex environment and the difficulty in extracting a particular feature that correlates with a particular event. As the number of cameras monitoring a given area increases, it will become vital to have systems capable of learning from the vast amounts of available data to identify any potential suspicious behavior. Then, the introduction of deep learning (DL) has brought new development directions for AD. In particular, DL models such as convolution neural networks (CNNs) and recurrent neural networks (RNNs) have achieved excellent performance dealing with AD tasks, as well as other challenging domains like image classification, object detection, and speech processing. In this review, we aim to present a comprehensive overview of those research methods using DL to address the AD problem. Firstly, different classifications of anomalies are introduced, and then the DL methods and architectures used for video AD are discussed and analyzed, respectively. The revised contributions have been categorized by the network type, architecture model, datasets, and performance metrics that are used to evaluate these methodologies. Moreover, several applications of video AD have been discussed. Finally, we outlined the challenges and future directions for further research in the field.