This chapter provides a comprehensive survey of the applications of semigroup theory across a diverse range of scientific and technical fields. Beginning with a historical and foundational overview, it establishes the semigroup as a powerful generalization of the group, uniquely suited to modelling systems where operations are associative but not necessarily invertible. The discussion is structured to demonstrate the unifying principles of semigroup applications, from their central role in the algebraic theory of automata and formal languages to their utility in modelling complex physical and biological systems. Specific areas of focus include the use of semigroups to represent state transitions in theoretical computer science, their function as a framework for solving partial differential equations, their critical role in describing the non-unitary dynamics of open quantum systems, and their utility in analyzing the evolution of Markov processes. The chapter concludes by highlighting the unifying theme of semigroups as a tool for analyzing systems with "non-invertible functions" and discusses emerging applications in bioinformatics and other fields.