Abstract: A real order system is a system described by a set of equations where state variables are associated with differential operators of real order. Stability analysis of any real order system is important in real order systems theory and applications. The idea of Lyapunov's theory is well-known in the theory of classical ordinary di erential systems. We will discuss new emerging Lyapunov theorems of real order systems that give stability of constant solutions to such systems. New extensions to the stability of any pair of non-trivial solutions to such systems will be addressed.