Reference Request for Optimal Transport I am reading through Villani's Optimal Transport: Old and New (2009) and I have a solid understanding of all the details presented in the first six chapters (essentially just Kantorovich duality, Wasserstein distances).  However, when I am attempting to read any modern papers in the subject, I find myself having a nearly complete lack of ability to understand most of what is going on.  I am looking for some suggestions for papers to read that might be accessible to someone with this fairly minimal background in OT.
In terms of my math background, it's not perfect by any stretch of the imagination.  I know probability/measure theory to the level of Billingsley and real analysis to the level of the first 10 chapters in Rudin.  Hopefully this is a good enough set of tools with which to understand deeper ideas in OT.  Thanks for any help you can provide!
 A: Well, I mean OT is a vast research area. I guess you will need to specify a little in which direction you wanna go. Besides, I personally wouldn't recommand Villani's opus magum as introductory literature. His other book 'topics in optimal transportation' or the one of Santambrogio are better for getting an overall idea of OT (personal opinion). But since you were asking for papers, I'd say the following four are amongst the 10 most influencial papers in OT of the last decades (which at the same time are not too hard to understand, I think. You may need however some background in PDE, stochastic analysis and/or Riemannian geometry (at least for the first 3), but that you will need in most of modern OT papers, anyway.):

Benamou, Brenier; A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem 
Jordan, Kinderlehrer, Otto; The Variational Formulation of the Fokker-Planck Equation
Otto; The geometry of dissipative evolution equations: the porous medium equation
Cuturi; Lightspeed Computation of Optimal Transport
