The explosive demand for digital data storage with larger storage capacity, higher reliability and fault tolerance, poses tremendous challenge on the storage industry. Distributed storage systems (DSS) are used to store massive amounts of data over a set of distributed storage nodes. Applications of DSS can be found in social networking, websites such as Facebook, cloud storage systems and video streaming website. Over time, storage nodes will leave the system, called node failures. A principal goal in DSS is the efficient repair of a failed node. tandard techniques for such a distributed way of storage include multiple backups (typically triple replication) or using erasure codes such as Reed-Solomon codes. But this techniques is not efficient.\\\\ \\emph{Locally repairable codes} (LRCs) are important, due to their applications in DSS. Let $\\mathcal{C}$ denote an $[n,k,d]$ linear code having block length $n$, dimension $k$ and minimum distance $d$. The $i$-th coordinate, $1\\leq i\\leq n$, of $\\mathcal{C}$ is said to have locality $r$, if the value at this coordinate can be repaired by accessing at most $r$ ($r \\ll k$) other coordinates and performing a linear computation. Further, if the $i$-th coordinate, $1\\leq i\\leq n$, can be repaired by accessing $r_i$ other coordinates, then $\\mathcal{C}$ is called an LRC with locality $r$.\\\\ In DSS, it is possible that multiple failed nodes occur at a time, and DSS must be able to repair as many failed nodes as possible. Basically, in case of multiple failed nodes, the repairing performance can heavily depend on the repairing strategy in use, say, repairing the failed nodes simultaneously or one by one. These two strategies are called parallel approach and sequential approach, respectively. In the parallel mode, the $t$ failed nodes can be repaired simultaneously, whereas in the sequential mode, the $u$ failed nodes are repaired one by one and the previously fixed failed nodes can be used in the next rounds of repairing. otentially, for a given LRC, the sequential mode can fix more failed nodes than the parallel mode ( In the other words $u\\geq t$); hence, for given parameters $r$ and $t=u$, the codes with sequential mode can potentially achieve higher rate and larger minimum distance than the parallel mode. In this thesis, we first consider LRC aiming at joint sequential-parallel repairing multiple failed nodes, and study the $(n,k,r,t,u)$-ELRCs (Exact locally repairable codes) which are $[n,k]$ linear codes with the property that any set of failed nodes of size at most $t$ can be simultaneously repaired in parallel mode, and each element of a set $E$ of failed nodes of size at most $u$ can be sequentially repaired by $r$ ($r k$) other coordinates. We present a method by which with a given parity-check matrix of an $(n,k,r,t,u)$-ELRC with minimum Hamming distance $d$, a new ELRC with minimum Hamming distance $2d$ and availability $t+1$ is constructed that can repair each set of failed nodes $E$ of size at most $2u + 1$ in sequential mode and this repair is done in at most $u-t+2$ steps. We construct a big family of LRCs by making use of orthogonal Latin rectangles, permutation cubes, back circulant Latin squares and some other combinatorial designs. Also, LRCs based on $t$-dimensional permutation $m$-cubes with block length $m^t$ are presented which have locality $r=2m-3$ and availability $t \\geq 3 $ in parallel mode, and also can repair any set of failed nodes of size up to $2^t-1$ in sequential mode in at most $t-1$ steps. Further, we show these codes are \\emph{overall local repair} with repair tolerance $2t$ in parallel mode. Further, we introduce some new clases of LRC, such as: Locally~repairable~code with arbitrary live node: The codes in this Sequential probabilistic repair: The codes in this Finally, we calculate the repair time (in the worst-case and average case) for some of the previously constructed codes in sequential mode.