Contents. 1. Preliminary information and auxialiary tools. 2. Identically distributed linear statistics based on a random sample. 3. Independant linear statistics. Modifications. Relativistic linear statistics. 4. Independant nonlinear statistics. Random linear statistics. 5. Regression problems connected with linear statistics. 6. Regression problems for n onlinear statistics. 7. Characterization of distributions throught the properties of admissibility and optimality of certain estimators. 8. Characterization of families of distributions admitting sufficient statistics. 9. Stability in characterization problems. 10. Characterization of random vectors with linear structure. 11. Polynomial statistics of a normal sample. 12. Characterizations of sequential estimation plans. 13. Other characteristic properties of distributions. 14. Unsolved problems. Addendum A. Addendum B.