A problem of third order approximations in limit theorems, especially in the central limit theorem, is discussed. Two opposite types of estimates, i.e. “real” and “formal” estimates are introduced. As an example of a “formal” estimate is given the Berry-Esseén estimate for the i.i.d. case with the theoretically minimal possible constant \((3+\sqrt{10})/6\sqrt{2\pi}\). It is called “formal” because for many important distributions and n not extremely large the obtained estimate is by 2 or 3 orders of magnitude worse than the true distance between the distributions.
The aim of the paper is not to provide new results but to discuss problems, difficulties and new ideas in this area. The author gives a review of the metrics in the space of probability measures and their properties and usefulness in the considered problems. In the last section, he proposes a method of getting “real” estimates exemplified again by the i.i.d. case. By some numerical analysis it is shown that this method gives much better estimates.