Fractional repetition (FR) codes is a family of codes for distributed storage systems that allow for uncoded exact repairs having the minimum repair bandwidth. However, in contrast to minimum bandwidth regenerating (MBR) codes, where a random set of a certain size of available nodes is used for a node repair, the repairs with FR codes are table based. This usually allows to store more data compared to MBR codes. In this work, we consider bounds on the fractional repetition capacity, which is the maximum amount of data that can be stored using an FR code. Optimal FR codes which attain these bounds are presented. The constructions of these FR codes are based on combinatorial designs and on families of regular and biregular graphs. These constructions of FR codes for given parameters raise some interesting questions in graph theory. These questions and some of their solutions are discussed in this paper. In addition, based on a connection between FR codes and batch codes, we propose a new family of codes for DSS, namely fractional repetition batch codes, which have the properties of batch codes and FR codes simultaneously. These are the first codes for DSS which allow for uncoded efficient exact repairs and load balancing which can be performed by several users in parallel. Other concepts related to FR codes are also discussed.