Image customization has been extensively studied in text-to-image (T2I) diffusion models, leading to impressive outcomes and applications. With the emergence of text-to-video (T2V) diffusion models, its temporal counterpart, motion customization, has not yet been well investigated. To address the challenge of one-shot motion customization, we propose Customize-A-Video that models the motion from a single reference video and adapting it to new subjects and scenes with both spatial and temporal varieties. It leverages low-rank adaptation (LoRA) on temporal attention layers to tailor the pre-trained T2V diffusion model for specific motion modeling from the reference videos. To disentangle the spatial and temporal information during the training pipeline, we introduce a novel concept of appearance absorbers that detach the original appearance from the single reference video prior to motion learning. The proposed modules are trained in a staged pipeline and inferred in a plug-and-play fashion, enabling easy extensions of our method to various downstream tasks such as custom video generation and editing, video appearance customization and multiple motion combination.