This thesis deals with the asymptotic and oscillatory behaviour of the solutions of certain differential and difference equations. It mainly consists of three parts. The first part is to study the asymptotic behaviour of certain differential equations. The second part is to look for oscillatory criteria for certain nonlinear neutral differential equations. And, the third part is to establish new criteria for a class of nonlinear neutral difference equations of any order with continuous variable and another type of higher even order nonlinear neutral difference equations to be oscillatory.