•An overview understanding of local flexibility markets (LFMs) is provided.•A comprehensive survey on formulations and clearing methods of LFMs are conducted.•Studies for solving operational issues of distribution systems are reviewed.•Deepen the understanding of LFMs in helping the operations of distribution systems. 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 provide trading platforms for local participants, including distribution system operators and aggregators. Starting with a survey on the concepts and definitions of local flexibility markets, we summarize the key elements, technologies and participants of local flexibility markets. Afterwards the formulations of local flexibility markets and the solution methods for market clearing are reviewed, based on which future research directions and challenges are suggested in the last section of this paper.