This article deals with the introduction of robust control theory which begins at the level of basic systems theory and ends having introduced the issues and machinery of current active research. The main aim is to provide the control-theoretic background required to read and contribute to the researchers. The research literature in robust control theory is vast and so we have not attempted to cover all topics, but have instead selected those that we believe are central and most effectively form a launching point for further study of the field. In particular, this paper focuses around the study concerning the robust stabilization of various dynamical systems with the aid of Lyapunov stability theory and linear matrix inequality (LMI) techniques.