The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp and fuzzy multisets have found numerous applications but still the basic operations are based on functions min and max. It is straightforward to replace these functions in the fuzzy part of fuzzy multisets; however, it is not as easy but is feasible to do the same with the multisets and the “crisp” part of fuzzy multisets. The new mathematical structures are called triangular multisets and triangular fuzzy multisets, respectively. The aim is to facilitate the processing of certain data sets so they can be used in multi-criteria decision making and computing.