Anomaly Detection is a relevant problem in numerous real-world applications, especially when dealing with images. However, little attention has been paid to the issue of changes over time in the input data distribution, which may cause a significant decrease in performance. In this study, we investigate the problem of Pixel-Level Anomaly Detection in the Continual Learning setting, where new data arrives over time and the goal is to perform well on new and old data. We implement several state-of-the-art techniques to solve the Anomaly Detection problem in the classic setting and adapt them to work in the Continual Learning setting. To validate the approaches, we use a real-world dataset of images with pixel-based anomalies to provide a reliable benchmark and serve as a foundation for further advancements in the field. We provide a comprehensive analysis, discussing which Anomaly Detection methods and which families of approaches seem more suitable for the Continual Learning setting.