In this chapter, the fundamentals of the theory of comparison systems described by state-space equations are presented. In Sect. 4.1, the comparison principle is defined and it is shown that Lyapunov’s direct method can be viewed as a method of construction of a stable monotone v-comparison system. In the following, two sections, v-comparison systems for continuous-time and discrete-time systems are presented. Section 4.4 deals with the problem of existence of converse l-comparison systems. Finally, in Sect. 4.5, the notion of polyhedral comparison is developed and applied to affine, linear, and Lur’e systems.