In this thesis, different classes of linear second-order singularly perturbed ordinary differential equations and differential-difference equations, both with or without turning points are considered. For the numerical treatment of these problems two methods namely, initial value method and parameter uniform hybrid finite difference method, are proposed. Apart from the construction of methods, a detailed theory for their convergence and error estimates are also presented. Extensive numerical experiments are also carried out to support the theoretical results. In cases of the singularly perturbed differential-difference equations, the effects of the small shift parameters on the layer behavior of the solutions are also examined. The initial value method which is an asymptoticnumerical approach in its nature is constructed to solve singularly perturbed differential-difference equations containing small shift parameters and a coupled system of second-order singularly perturbed ordinary differential equations, whenever the solution of these problems exhibits a single boundary layer. It is observed that this method is simple to apply, very easy to implement on a computer, and offers a relatively simple tool for obtaining an approximate solution. Besides, the present method has better efficiency than some of the existing methods in the literature. On the other hand, a parameter uniform hybrid finite difference scheme on a layer resolving piecewiseuniform Shishkin mesh is employed for approximating the solutions of second-order singularly perturbed ordinary differential equations and differential-difference equations with a turning point, when the solutions of these problems exhibit exponential boundary layers at both ends of the domain. It is proved that the method gives an almost second order e-uniform convergence and also has a superior order of convergence than some of the existing methods.