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How to make a segmentation model efficiently adapt to a specific video and to online target appearance variations are fundamentally crucial issues in the field of video object segmentation. In this work, a graph memory network is developed to address the novel idea of "learning to update the segmentation model". Specifically, we exploit an episodic memory network, organized as a fully connected graph, to store frames as nodes and capture cross-frame correlations by edges. Further, learnable controllers are embedded to ease memory reading and writing, as well as maintain a fixed memory scale. The structured, external memory design enables our model to comprehensively mine and quickly store new knowledge, even with limited visual information, and the differentiable memory controllers slowly learn an abstract method for storing useful representations in the memory and how to later use these representations for prediction, via gradient descent. In addition, the proposed graph memory network yields a neat yet principled framework, which can generalize well both one-shot and zero-shot video object segmentation tasks. Extensive experiments on four challenging benchmark datasets verify that our graph memory network is able to facilitate the adaptation of the segmentation network for case-by-case video object segmentation.