In this thesis, parameter-uniform numerical methods for certain classes of singularly perturbed differential equations with two small parameters are studied. We initially consider a class of two-parameter ordinary differential equations Parameter explicit bounds on the solution and its derivatives are derived. The solution is decomposed into a sum of regular and singular components and based on this decomposition we construct a numerical algorithm consisting of an upwind finite difference operator and an appropriately chosen piecewise-umform mesh. Parameter-uniform convergence of the numerical approximations is established Some numerical results are given to illustrate this convergence. Two-parameter parabolic and elliptic partial differential equations are considered. We derive parameter explicit bounds on the solutions and their derivatives for both problems, these bounds are analogous to those obtained for the ordinary differential equation. The solutions are decomposed into a sum of regular and singular components but for both problems this decomposition differs from that for the ordinary differential equation. In both cases a numerical algorithm based on an upwind finite difference operator and an appropriate piecewise-umform mesh is constructed. In the case of the parabolic problem, parameter-uniform error bounds for the numerical approximations are established and numerical results illustrating this convergence are given With the elliptic. problem, we show that, given certain assumptions and conjectures our numerical method is parameteruniform