First I will introduce the strong and the relaxed formulations of the multi-marginal optimal transportation problem and I will compare them with the 2-marginal version. I will then explain which are the key tools which allowed the solution in the 2-marginal case and are missing in the multi-marginal problem. For particular costs advancement have been made and, for sufficiently general costs, partial solutions have been given. This positive results will be discussed in the second part of the expository talk. Finally some attention will be dedicated to costs which appeared in applications related to the theme of the meeting.