In the first part of the thesis, we derive high-order finite difference schemes for solving steady-state convection-diffusion problems. The schemes exhibit higher order accuracy at the grid points yet utilize only compact stencils. The derivation is based on using the governing differential equation to approximate the leading truncation error terms in the central difference approximation. A comparison with the standard central and upwind difference schemes is made. Numerical experiments in one- and two-dimensional spaces are conducted to show the high-order accuracy of the HOC schemes, as well as their tendency to suppress false oscillations, In the second part of the thesis, we consider time-dependent problems. We derive a high-order compact alternating direction implicit (ADI) method for solving two- and three-dimensional unsteady convection-diffusion problems. The method is fourth-order in space and second-order in time. It permits multiple uses of the one dimensional tridiagonal algorithm with a considerable saving in computing time, and results in a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with well-known existing ADI methods.