The difficulty in studying the stability of ordinary arid delay differential equations is that there is no general procedure one can depend on for this purpose. The following is some of what have been achieved in this thesis 1- To present some of the more important methods of studying the stability of ordinary and delay differential equations; formulating examples to show the details of method. Merits and drawbacks for each method are stated.the procedure of each 2- Writing algorithms with computer programs for some of the described methods and using them to study a number of examples to show its efficiency. Developing some methods, and a new technique is proposed for studying the stability of delay differential equations. 4-Introducing some applications of ordinary and delay differential equations and sV-jdying their stability by using he methods that are described in this thesis with new provement in this domain.