In the first part of the talk we will look at the optimal transport problem from two different approaches, analyzing some of their strengths, weaknesses and sketching the intuitions behind each one. Such approaches will be: the classic Monge-Kantorovich formulation and the Eulerian point of view given by the Benamou-Brenier formulation. In the second part the classical Schrödinger problem will be presented, and we will see how this problem from statistichal mechanics can be seen as a regularization of the optimal transport problem.