Classical multi-marginal problems ask whether a collection of lower-order marginals can arise from a common probability measure, and variational problems under such marginal constraints play an important role in optimal transport and density functional theory. Their quantum analogue is the $m$-representability, or quantum marginal, problem : determine whether prescribed reduced density operators arise as the marginals of a common $m$-particle quantum state. In particular, the fermionic $m$-representability problem–characterizing which reduced density matrices arise from an $m$-electron quantum state–remains a major outstanding problem in electronic structure theory and quantum chemistry. The goal of this introductory talk is to formulate these problems and describe some of the most accessible open questions.