The performance of multi-domain image-to-image translation has been significantly improved by recent progress in deep generative models. Existing approaches can use a unified model to achieve translations between all the visual domains. However, their outcomes are far from satisfying when there are large domain variations. In this paper, we reveal that improving the sample selection strategy is an effective solution. To select informative samples, we dynamically estimate sample importance during the training of Generative Adversarial Networks, presenting Informative Sample Mining Network. We theoretically analyze the relationship between the sample importance and the prediction of the global optimal discriminator. Then a practical importance estimation function for general conditions is derived. Furthermore, we propose a novel multi-stage sample training scheme to reduce sample hardness while preserving sample informativeness. Extensive experiments on a wide range of specific image-to-image translation tasks are conducted, and the results demonstrate our superiority over current state-of-the-art methods.