Data storage has grown such that distributed storage over a number of systems is now commonplace. This has given rise to an increase in the complexity of ensuring data loss does not occur, particularly where failure is due to the failure of individual nodes within the storage system. Redundancy was the main tool to combat this, but with huge increases in data, minimization of the overhead associated with this technique caused major concern. In a large data center, a third concern arose, namely the need for efficient recovery from the failure of a single storage unit.In this monograph, the authors give a comprehensive overview of the role of differing types of codes in addressing the issues in large distributed storage systems. They introduce the reader to regenerative codes, locally recoverable codes and locally regenerative codes; the three main classes of codes used in such systems. They give an exhaustive overview of how these codes were created, their uses and the developments and improvements of the codes in the last decade.This in-depth review gives the reader an accessible and complete overview of the modern codes used in distributed storage systems today. It is a one-stop source for students, researchers and practitioners working on any such system.