Glaucoma is one of the leading causes of irreversible blindness. Segmentation of optic disc (OD) and optic cup (OC) on fundus images is a crucial step in glaucoma screening. Although many deep learning models have been constructed for this task, it remains challenging to train an OD/OC segmentation model that could be deployed successfully to different healthcare centers. The difficulties mainly comes from the domain shift issue, i.e., the fundus images collected at these centers usually vary greatly in the tone, contrast, and brightness. To address this issue, in this paper, we propose a novel unsupervised domain adaptation (UDA) method called Reconstruction-driven Dynamic Refinement Network (RDR-Net), where we employ a due-path segmentation backbone for simultaneous edge detection and region prediction and design three modules to alleviate the domain gap. The reconstruction alignment (RA) module uses a variational auto-encoder (VAE) to reconstruct the input image and thus boosts the image representation ability of the network in a self-supervised way. It also uses a style-consistency constraint to force the network to retain more domain-invariant information. The low-level feature refinement (LFR) module employs input-specific dynamic convolutions to suppress the domain-variant information in the obtained low-level features. The prediction-map alignment (PMA) module elaborates the entropy-driven adversarial learning to encourage the network to generate source-like boundaries and regions. We evaluated our RDR-Net against state-of-the-art solutions on four public fundus image datasets. Our results indicate that RDR-Net is superior to competing models in both segmentation performance and generalization ability