In mathematics, a nonmeasurable set is one for which the "volume" cannot be assigned. This "volume" can be understood differently depending on the structure in which we search sucha a set. In fact, from the very beginning, the notation of nonmeasurable set was a source of considerable controversy.CONTENTSIntroductionI1. Preliminaries2. nonmeasurable setsII3. Kuratowski partitions4. On Kuratowski partitions in tree structures5. On Kuratowski partitions in Ellentuck topology6. Ideals associated with Kuratowski partitions7. Kuratowski partitions in Baire spaces8. Kuratowski partitions and game theory9. Kuratowski partitions in complete metric spaces10. An example of a metric space without Kuratowski partitionsIII11. The generalization of Louveau-Simpson Theorem12. On the equivalences of Gitik-Shelak Theorem13. The generalization of Halpern-Lauchli TheoremIV14. {Partitions and point-finite covers in Baire spaces15. Nonmeasurable unions for point-finite families16. On the existence of measurable selectorsBibliographyIndex