Locally repairable convolutional codes (LRCCs) for distributed storage systems (DSSs) are introduced in this work. They enable local repair, for a single node erasure (or more generally, $ \partial - 1 $ erasures per local group), and sliding-window global repair, which can correct erasure patterns with up to $ {\rm d}^c_j - 1 $ erasures in every window of $ j+1 $ consecutive blocks of $ n $ nodes, where $ {\rm d}^c_j $ is the $ j $th column distance of the code. The parameter $ j $ can be adjusted, for a fixed LRCC, according to different catastrophic erasure patterns, requiring only to contact $ n(j+1) - {\rm d}^c_j + 1 $ nodes, plus less than $ μn $ other nodes, in the storage system, where $ μ$ is the memory of the code. A Singleton-type bound is provided for $ {\rm d}^c_j $. If it attains such a bound, an LRCC can correct the same number of catastrophic erasures in a window of length $ n(j+1) $ as an optimal locally repairable block code of the same rate and locality, and with block length $ n(j+1) $. In addition, the LRCC is able to perform the flexible and somehow local sliding-window repair by adjusting $ j $. Furthermore, by adjusting and/or sliding the window, the LRCC can potentially correct more erasures in the original window of $ n(j+1) $ nodes than an optimal locally repairable block code of the same rate and locality, and length $ n(j+1) $. Finally, the concept of partial maximum distance profile (partial MDP) codes is introduced. Partial MDP codes can correct all information-theoretically correctable erasure patterns for a given locality, local distance and information rate. An explicit construction of partial MDP codes whose column distances attain the provided Singleton-type bound, up to certain parameter $ j=L $, is obtained based on known maximum sum-rank distance convolutional codes.