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In unsupervised person Re-ID, peer-teaching strategy leveraging two networks to facilitate training has been proven to be an effective method to deal with the pseudo label noise. However, training two networks with a set of noisy pseudo labels reduces the complementarity of the two networks and results in label noise accumulation. To handle this issue, this paper proposes a novel Dual Clustering Co-teaching (DCCT) approach. DCCT mainly exploits the features extracted by two networks to generate two sets of pseudo labels separately by clustering with different parameters. Each network is trained with the pseudo labels generated by its peer network, which can increase the complementarity of the two networks to reduce the impact of noises. Furthermore, we propose dual clustering with dynamic parameters (DCDP) to make the network adaptive and robust to dynamically changing clustering parameters. Moreover, Consistent Sample Mining (CSM) is proposed to find the samples with unchanged pseudo labels during training for potential noisy sample removal. Extensive experiments demonstrate the effectiveness of the proposed method, which outperforms the state-of-the-art unsupervised person Re-ID methods by a considerable margin and surpasses most methods utilizing camera information.