This dissertation presents a concise history of the uses of multisets in various disguised forms which eventually led to a formalization of multiset theory. We present a comprehensive study of fundamentals of multisets and their applications. We study max-plus algebra and by exploiting the notion of compatibility relation (reflexive and symmetric), we develop a truncated symmetrized max-plus algebra called minimal Smax or Smax, and discuss its application. Finally, we outline some multiset algebras developed in the literature. We explicate the notion of multiset space, introduce two new...