We consider the problem of video caching in an edge network consisting of a set of small-cell base stations (SBS) that can share content among themselves over a high-capacity short-delay local network, or fetch the videos from a remote server over a long-delay connection. Even though the problem of minimizing the overall video playout delay in our framework is NP-hard, we develop CLoSER, an algorithm that can efficiently compute a solution that is close to the optimal, where the degree of sub-optimality depends on the worst case video-to-cache size ratio. In comparison with related prior work on video caching and streaming, CLoSER specifically focuses on the benefits of local sharing of the initial portion of the video content in reducing the video playout delay, and provides strong optimality guarantees with a low-complexity algorithm. We extend CLoSER to an online setting where the video popularities are not known a priori but are estimated over time through a limited amount of periodic information sharing between SBSs. With such online video popularity estimation, a distributed implementation of CLoSER requires zero explicit coordination between the SBSs and runs in O(NK+KlogK) time, where N is the number of SBSs (caches) and K the maximum number of videos. We carry out simulations using YouTube and Netflix video request traces, as well as synthesized traces with the same marginal distributions as the traces but varying degree of temporal correlations, and demonstrate that our algorithm uses the SBS caches effectively to reduce the video delivery delay and conserve the remote server’s bandwidth. We also show that it outperforms two other reference caching methods adapted to our system setting, over a wide range of remote-to-local bandwidth ratios. Further, we show how CLoSER extends to the scenario where each video may need to be cached in multiple bit-rates, as in Available Bit Rate (ABR) streaming.