In recent years numerous definitions of stability having a more quantitative nature than most classical stability definitions have been presented by various investigators. These definitions involve known specific bounds on solutions of tho differential equation in question. In this paper a now type of stability (‘set stability’) is introduced which generalizes upon and in some respects unifies those recent endeavours in ‘ quantitative stability theory ’. The usual ‘Liapunov technique’ is employed in obtaining sufficient conditions for the various forms of set stability.