Deep learning-based industrial anomaly detection models have achieved remarkably high accuracy on commonly used benchmark datasets. However, the robustness of those models may not be satisfactory due to the existence of adversarial examples, which pose significant threats to the practical deployment of deep anomaly detectors. Recently, it has been shown that diffusion models can be used to purify the adversarial noises and thus build a robust classifier against adversarial attacks. Unfortunately, we found that naively applying this strategy in anomaly detection (i.e., placing a purifier before an anomaly detector) will suffer from a high anomaly miss rate since the purifying process can easily remove both the anomaly signal and the adversarial perturbations, causing the later anomaly detector failed to detect anomalies. To tackle this issue, we explore the possibility of performing anomaly detection and adversarial purification simultaneously. We propose a simple yet effective adversarially robust anomaly detection method, \textit{AdvRAD}, that allows the diffusion model to act both as an anomaly detector and adversarial purifier. We also extend our proposed method for certified robustness to $l_2$ norm bounded perturbations. Through extensive experiments, we show that our proposed method exhibits outstanding (certified) adversarial robustness while also maintaining equally strong anomaly detection performance on par with the state-of-the-art methods on industrial anomaly detection benchmark datasets.