Publisher Summary This chapter discusses the mappings of spaces. Whenever an arbitrary mapping is considered, it is assumed that this mapping is continuous. The universality problem for the classes consisting mappings is studied. For this purpose, the main construction of Containing Spaces is applied separately to the class consisting the domains and to the class consisting the ranges of all the mappings of the considered class. Classes consisting mappings with different ranges are considered in the chapter. The notion of a saturated class is given, and the intersection and the universality properties of such classes are proved. The chapter also proves the following results: (1) the class of the domains and the class of the ranges of all elements of a saturated class of mappings are saturated classes of spaces and (2) the class of all (open) mappings whose domains and ranges belong to a given saturated classes of spaces is saturated.