In this thesis the solutions of the two-dimensional (2D) and three-dimensional (3D) lid-driven cavity problem are obtained by solving the steady Navier-Stokes equations at high Reynolds numbers. In 2D, we use the streamfunction-vorticity formulation to solve the problem in a square domain. A numerical method is employed to discretize the problem in the x and y directions with a spectral collocation method. The problem is coded in the MATLAB programming environment. Solutions at high Reynolds numbers are obtained up to $Re=25000$ on a fine grid of 131 * 131. The same method is also used to obtain the numerical solutions for the steady separated corner flow at high Reynolds numbers are generated using a for various domain sizes, at various Reynolds number which are much higher than those obtained by other researchers.Finally, the numerical solutions for the three-dimensional lid-driven cavity problem are obtained by solving the velocity-vorticity formulation of the Navier-Stokes equations for various Reynolds numbers. A spectral collocation method is employed to discretize the y and z directions and finite difference method is used to discretize the x direction. Numerical solutions are obtained for Reynolds number up to 200.