This book studies the basic problems like observability, controllability, stability, stabilizability and optimal control for dynamical systems represented by ordinary differential equations in a finite dimensional Euclidean space. The problems are also considered for nonlinear dynamical systems. The contents of the book are organized to serve as an introductory level textbook helping to understand the basic ingredients of control theory. A good number of examples are provided to illustrate the concepts. The prerequisites are elementary courses in analysis, differential equations and the theory of matrices.