This chapter thoroughly explores the Laplace transform, a critical tool for analyzing complex systems in engineering and applied sciences. It opens with a historical background, establishing the transform’s role in simplifying differential equations and dynamic system analysis. The chapter defines the Laplace transform and explains the conditions for its existence and uniqueness, ensuring its applicability across diverse functions. Key properties like linearity, shifting, and scaling are detailed, offering efficient methods for manipulating transformed functions. The coverage of the inverse Laplace transform follows, with techniques like partial fraction decomposition and contour integration for reconstructing original functions. Heaviside’s expansion theorem enhances inversion methods, while differentiation and integration in the Laplace domain allow for refined modeling. Initial and final value theorems provide insights into system behavior at endpoints, with applications in system modeling underscoring the transform’s engineering utility. The chapter concludes with the bilateral Laplace transform and standard function examples, equipping readers with a solid grounding in both theory and practical applications of the Laplace transform.