The Clean Energy Package aims to transit towards cleaner energy in the European Union and requires market reforms towards small-scale flexibility provision and the creation of local flexibility markets. However, the progress in many Member States is slow. While several Member States have initiated a reform of their existing energy market structures, only few have started the creation of local flexibility markets. This paper provides an overview of emerging flexibility markets in the EU. This paper also provides an analysis of important elements of local flexibility markets based on literature review and assessments made in the Horizon 2020 project X-FLEX. Seven key characteristics of local flexibility markets have been retained from this literature review. Small-scale flexibility is still a relatively new concept, and a lot of barriers prevent it to enter the markets. Two main types of gaps were identified: (i) marketrelated gaps, such as aggregation rules that are not open enough, and (ii) operational gaps, such as the need for the large-scale smart metering infrastructure rollout and a large amount of data streams. The operation of local flexibility markets should depend on the network conditions they are applied to, as the conditions are different from one network to another. For this reason, the need for local flexibility markets has to be assessed depending on the problems the network is currently facing or will be facing in the future due to foreseen increased amount of distributed generation sources and electrified demand. A detailed analysis in terms of flexibility provision for the four demonstration sites in three European Member States participating in the X-FLEX project is presented in this paper, along with recommendations and activities for local flexibility market development performed in these demonstration sites.