On the basis of K. Kuratowski’s definitions of a topological structure with a closure or interior operator, the concept of a modal topological structure (MTS) with one of these operators was introduced by the author. This new structure was illustrated with examples with intuitionistic fuzzy topological operators from both examples, and for this reason, these structures were named intuitionistic fuzzy MTSs (IFMTSs). In a series of papers, the author introduced some modifications and extensions to the IFMTSs, e.g., intuitionistic fuzzy temporal topological structures, intuitionistic fuzzy level topological structures and others, and intuitionistic fuzzy multi modal topological structures and others. In the present paper, four new examples of intuitionistic fuzzy multi modal topological structures are given. On their base, the concepts of a modal multi-topological structure and of a multi-modal multi-topological structure are introduced and illustrated with examples from the area of the intuitionistic fuzzy sets—intuitionistic fuzzy modal multi-topological structure with a closure or an interior operator; and intuitionistic fuzzy multi-modal multi-topological structure with one of these operators. Two intuitionistic fuzzy topological operators are defined. Their basic properties are studied and they are used in the new structures.