The objective of this chapter, is the presentation of the Laplace transform, a fundamental transform in the field of signal processing, that allows the description of systems in the complex frequency domain. Starting with the description of the defining equation and the conditions of convergence of this transform, the statement of the existence theorem and the solution of illustrative examples of its calculation, the chapter proceeds with the proof of the properties of the Laplace transform, the presentation of the initial and final value theorems, as well as the introduction of the very important concepts of the poles and zeros, associated with each rational function of complex variable. Then, the calculation of the inverse Laplace transform is presented in detail, via the application of the three related fundamental methods, namely the method of integration in the complex plane, the method of power series expansion, as well as the method of partial fraction expansion. The Heaviside’s theorem, is also presented. The chapter concludes with the application of the Laplace transform for solving ordinary differential equations, as well as, for the description of continuous first and second-order systems.