In 1931 E. Schrödinger addressed the following question: what is the most likely evolution of a cloud of i.i.d. Brownian particles, conditionally on the observation of their initial and final configurations? In the beginning of the talk, we will discuss how this problem (the Schrödinger problem) admits several variational formulations, related to large deviations, optimal transport, and control theory. In particular, leveraging on the duality between Fokker-Planck and Hamilton-Jacobi-Bellman equations, we will be able to obtain duality formulas. However, Schrödinger's original thought experiment can be formulated for particles other than Brownian ones. We will see that (with different degrees of formalism) the primal formulations of the Schrödinger problem can be naturally extended to model various particle systems where interaction is allowed. We will notably focus on the so-called "Mean-Field Schrödinger problem", remarking that the picture is now more complicated: the Fokker-Planck-HJB duality argument leads indeed to a Hamilton-Jacobi equation on the Wasserstein space. The final part of the talk will thus be devoted to the study of viscosity solutions to such PDE, presenting recent results obtained in collaboration with G. Conforti, R. Kraaij, and D. Tonon.