In this talk we focus on matching markets under preferences. In particular, we investigate a scenario where each agent in a market has an ordered preference list over some other agents and our task is to find an optimal matching of the agents so that it represents a certain global social welfare. A matching M is popular if there is no matching M' such that the vertices that prefer M' to M outnumber those that prefer M to M'. In this talk, we will summarize known results about the existence and structure of popular matchings under various assumptions such as bipartite and general graphs, strictly ordered lists and lists containing ties. We will also pose several open questions of this quickly growing field.