We propose a self-supervised spatio-temporal matching method, coined Motion-Aware Mask Propagation (MAMP), for video object segmentation. MAMP leverages the frame reconstruction task for training without the need for annotations. During inference, MAMP extracts high-resolution features from each frame to build a memory bank from the features as well as the predicted masks of selected past frames. MAMP then propagates the masks from the memory bank to subsequent frames according to our proposed motion-aware spatio-temporal matching module to handle fast motion and long-term matching scenarios. Evaluation on DAVIS-2017 and YouTube-VOS datasets show that MAMP achieves state-of-the-art performance with stronger generalization ability compared to existing self-supervised methods, i.e., 4.2% higher mean J&F on DAVIS-2017 and 4.85% higher mean J&F on the unseen categories of YouTube-VOS than the nearest competitor. Moreover, MAMP performs at par with many supervised video object segmentation methods. Our code is available at: https://github.com/bo-miao/MAMP.