Anomaly detection on multivariate time series (MTS) is an important research topic in data mining, which has a wide range of applications in information technology, financial management, manufacturing system, and so on. However, the state-of-the-art unsupervised deep learning models for MTS anomaly detection are vulnerable to noise and have poor performance on the training data containing anomalies. In this article, we propose a novel Self-Training based Anomaly Detection with Generative Adversarial Network (GAN) model called STAD-GAN to address the practical challenge. The STAD-GAN model consists of a generator-discriminator structure for adversarial learning and a neural network classifier for anomaly classification. The generator is learned to capture the normal data distribution, and the discriminator is learned to amplify the reconstruction error of abnormal data for better recognition. The proposed model is optimized with a self-training teacher-student framework, where a teacher model generates reliable high-quality pseudo-labels to train a student model iteratively with a refined dataset so that the performance of the anomaly classifier can be gradually improved. Extensive experiments based on six open MTS datasets show that STAD-GAN is robust to noise and achieves significant performance improvement compared to the state-of-the-art.