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With high penetration of renewable generation and distributed energy resources, distribution systems are facing new operational challenges due to their intermittency and uncertainty. To cope with these challenges, distribution system operators are seeking for market tools to enable more active system management and control using flexibility. Local flexibility markets provide opportunities to trade flexibility among distribution system operators and other participants (e.g., aggregators) in an economically efficient way. Studies have been carried out to define the concepts, design the mechanisms and formulate the market clearing methods of local flexibility markets in the literature. Therefore, there is a need to classify and organize the literature on potential designs and market clearing methods of local flexibility markets. This paper reviews local flexibility markets, which are currently being discussed and designed to …