The book is devoted to universality problems. A new approach to these problems is given using some specific spaces. Since the construction of these specific spaces is set-theoretical, the given theory can be applied to different topics of Topology such as: universal mappings, dimension theory, action of groups, inverse spectra, isometrical embeddings, and so on. Universal spaces Universal mappings Dimension theory Actions of groups Isometric Universal SpacesPreface. Introductory remarks.Chapter 1. The construction of Containing Spaces.Chapter 2. Saturated classes.Chapter 3. Dimension-like functions.Chapter 4. Saturated classes of spaces with structure.Chapter 5. Completely regular and compact spaces.Chapter 6. Saturated classes of mappings.Chapter 7. Actions of groups.Chapter 8. Containing Spaces and factorizing T-spectra.Chapter 9. Isometries and universality.Chapter 10. Concluding remarks and open problems.brBibliography. Index of symbols. Subject index. AudienceDescription: Researchers in Topology, postgraduate students.