Morita Equivalence of C*-Crossed Products by Inverse Semigroup Actions and Partial Actions Abstract: This article explores two concepts of Morita equivalence related to twisted inverse semigroup actions and discrete twisted partial actions. The connection between inverse semigroup actions and partial actions facilitates the notion of Morita equivalence for discrete twisted partial actions. In particular, the paper shows that Morita equivalent actions lead to Morita equivalent crossed products. 1. Introduction: Previous studies have examined Morita equivalence of group actions on C*-algebras. In this article, this concept is extended to Busby-Smith and Green type inverse semigroup actions. The correlation between inverse semigroup actions and partial actions makes it possible to identify the Morita equivalence for discrete twisted partial actions. The paper details discrete twisted partial crossed products and reaffirms that Morita equivalent twisted partial actions result in Morita equivalent crossed products. 2. Preliminaries: This section explains several key definitions to clarify terminology and notation. It discusses various concepts including B-module, right inner-product, and Hilbert B-module as well as B-valued inner product. The concept of a Hilbert A−B-bimodule is also presented, which is a bimodule that is a left Hilbert A-module and a right Hilbert B-module. 3. Morita Equivalent Twisted Actions: This section introduces the concept of a partial automorphism of the imprimitivity bimodule and Morita equivalent twisted actions. The definition of a Busby-Smith twisted action of an inverse semigroup on a C*-algebra is recalled. Covariant representation of a twisted action is discussed. Two Busby-Smith twisted actions are said to be Morita equivalent if an imprimitivity bimodule and a specific map exist. The research for the article was conducted at Arizona State University under Professor John Quigg's guidance and supported by a grant from the National Science Foundation. It introduces essential updates on Morita equivalence of twisted inverse semigroup actions and discrete twisted partial actions, with potential for further advancements in mathematical theories.