I will present joint results with H.-M. Nguyen concerning the study of the Jacobian determinant of maps from R^N into R^N (and also S^N into S^N). New estimates in the Wasserstein metric illuminate classical results of Y. Reshetnyak and J.Ball concerning the behavior of the distributional Jacobian under weak convergence e.g. in W^{1,N}. I will also discuss the optimal Sobolev space where one can define a concept of weak Jacobian.