Nuclei segmentation is a key technique for automatic pathological screening. Although many methods have been proposed, it remains a challenge because of numerous nuclei clusters, high variability of object appearances and complex backgrounds. To address these issues, we propose a novel multi-task region-enhanced nuclei segmentation network (REU-Net). It stacks three U-shaped structures by combining serial and parallel approaches to construct a multi-task architecture. The model employs two auxiliary tasks, i.e., contour extraction and rough segmentation to help the main task of fine segmentation. The saliency regions are enhanced by the prediction results of the auxiliary tasks, and the enhanced images are further segmented through the main task. In addition, the spatial and texture features in auxiliary tasks are aggregated by attention gates, helping the main task to refine the details of nuclei and contours. Extensive experiments are conducted to evaluate the proposed method qualitatively and quantitatively. Experimental results show that REU-Net outperforms the state-of-the-art methods on HUSTS, MoNuSeg, CoNSep and CPM-17 datasets.