  The Skorokhod Embedding Problem (SEP) is one of the classical problems in the
study of stochastic processes, with applications in many different fields (cf.~
the surveys \cite{Ob04,Ho11}). Many of these applications have natural
multi-marginal extensions leading to the \emph{(optimal) multi-marginal
Skorokhod problem} (MSEP). Some of the first papers to consider this problem
are \cite{Ho98b, BrHoRo01b, MaYo02}. However, this turns out to be difficult
using existing techniques: only recently a complete solution was be obtained in
\cite{CoObTo15} establishing an extension of the Root construction, while other
instances are only partially answered or remain wide open.
  In this paper, we extend the theory developed in \cite{BeCoHu14} to the
multi-marginal setup which is comparable to the extension of the optimal
transport problem to the multi-marginal optimal transport problem. As for the
one-marginal case, this viewpoint turns out to be very powerful. In particular,
we are able to show that all classical optimal embeddings have natural
multi-marginal counterparts. Notably these different constructions are linked
through a joint geometric structure and the classical solutions are recovered
as particular cases.
  Moreover, our results also have consequences for the study of the martingale
transport problem as well as the peacock problem.
