This book presents methods of analyzing essentially nonlinear autonomous systems of ordinary differential equations. The book is in two parts. The first discusses a method combining the Lyapunov method, the Poincare method of small parameter, and the method of averaging. The second part deals with the application of the theory of normal forms to autonomous systems of third-, fourth-, and sixth-order equations. Mechanical, physical and electromechanical systems are discussed as examples. The book is intended for specialists in applied mathematics, including undergraduate and postgraduate students.