In recent years, power systems have undergone changes in technology and definition of the associated stakeholders. With the increase in distributed renewable generation and small-to medium-sized consumers starting to actively participate on the supply side, a suitable incorporation of decentralized agents into the power system is required. A promising scheme to support this shift is given by local electricity markets. These provide an opportunity to extend the liberal wholesale markets for electrical power found in Europe and the United States to the communal level. Compared to these more established markets, local electricity markets, however, neither have few practical implementations nor standardized frameworks. In order to fill this research gap and classify the types of local electricity markets, the presented paper therefore starts with the challenges that these markets attempt to solve. This is then extended to an analysis of the theoretical and practical background with a focus on these derived challenges. The theoretical background is provided in the form of an introduction to state-of-the-art models and the associated literature, whereas the practical background is provided in form of a summary of ongoing and recent projects on local electricity markets. As a result, this paper presents a foundation for future research and projects attempting to approach the here presented challenges in distribution of generation, integration of demand response, decentralization of markets and legal and social issues via local electricity markets.