In modern power systems high penetration of renewable energy sources and decentralized paradigm are regarded as the path toward more sustainable electricity landscape. This includes distributed energy resources whose intermittency and uncertainty may cause issues to the system operators. Distribution system operators have an obligation to ensure secure and stable system operation. Hence, they seek the most efficient methods to deal with these challenges. Flexibility procurement is considered as one of the prerequisites for painless and successful integration of renewable sources. Furthermore, distribution-level flexibility markets are modeled and tested to trade flexibility locally, solve congestion issues and defer grid expansion. This paper surveys the ongoing research in the field of flexibility markets, its design, open questions and most promising research projects. The key stakeholders are identified, overview of the current trends in the power system and research initiatives are presented, accompanied with the dilemmas being discussed in the power systems community.