This chapter introduces the elements of stability theory for evolution equationsEvolution equation in abstract spaces. Special attention is paid to an analysis of exponential, finite-, and fixed-time stabilityFixed-time stability (FxTS)  being typical for homogeneous systems. Contrary to the finite-dimensional case, the finite/fixed-time stabilityFixed-time stability (FxTS) in the infinite-dimensional setting can be discovered even for linear systems. The Lyapunov functionFunction method is adapted for the analysis of stability and convergence rate of an abstract differential equation. Lyapunov theorems are obtained for various systems.