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Motion transfer aims to transfer the motion of a driving video to a source image. When there are considerable differences between object in the driving video and that in the source image, traditional single domain motion transfer approaches often produce notable artifacts; for example, the synthesized image may fail to preserve the human shape of the source image (cf . Fig. 1 (a)). To address this issue, in this work, we propose a Motion and Appearance Adaptation (MAA) approach for cross-domain motion transfer, in which we regularize the object in the synthesized image to capture the motion of the object in the driving frame, while still preserving the shape and appearance of the object in the source image. On one hand, considering the object shapes of the synthesized image and the driving frame might be different, we design a shape-invariant motion adaptation module that enforces the consistency of the angles of object parts in two images to capture the motion information. On the other hand, we introduce a structure-guided appearance consistency module designed to regularize the similarity between the corresponding patches of the synthesized image and the source image without affecting the learned motion in the synthesized image. Our proposed MAA model can be trained in an end-to-end manner with a cyclic reconstruction loss, and ultimately produces a satisfactory motion transfer result (cf . Fig. 1 (b)). We conduct extensive experiments on human dancing dataset Mixamo-Video to Fashion-Video and human face dataset Vox-Celeb to Cufs; on both of these, our MAA model outperforms existing methods both quantitatively and qualitatively.