This paper proposes a novel memory-based online video representation that is efficient, accurate and predictive. This is in contrast to prior works that often rely on computationally heavy 3D convolutions, ignore actual motion when aligning features over time, or operate in an off-line mode to utilize future frames. In particular, our memory (i) holds the feature representation, (ii) is spatially warped over time to compensate for observer and scene motions, (iii) can carry long-term information, and (iv) enables predicting feature representations in future frames. By exploring a variant that operates at multiple temporal scales, we efficiently learn across even longer time horizons. We apply our online framework to object detection in videos, obtaining a large 2.3 times speed-up and losing only 0.9% mAP on ImageNet-VID dataset, compared to prior works that even use future frames. Finally, we demonstrate the predictive property of our representation in two novel detection setups, where features are propagated over time to (i) significantly enhance a real-time detector by more than 10% mAP in a multi-threaded online setup and to (ii) anticipate objects in future frames.