This volume, which coincides with the centennial anniversary of the publication of the celebrated monograph "The General Problem of the Stability Motion" by A.M. Liapunov, reviews the current state of the art of the theory and applications of the Liapunov methods. The text contains an introduction and four chapters. Chapter 2 presents some general results in stability theory. The remaining chapters deal with applications in power engineering, chemical engineering, and in non-engineering fields such as economics and in the modelling of interacting species. The diversity of mathematical tools employed, and the described approach to mathematical modelling provide considerations for applications in many other fields. The text is suitable for mathematicians and engineers whose work involves the study and applications of stability theory in systems.