Prandtl in 1904 discovered that the flow of a fluid over a thin obstacle can be adequately represented by an approximate set of equations, much simpler than the complex Navier-Stokes equations which govern the motion of fluid. A particularly simple for of these equations, for the two-dimensional steady flow of a fluid past a flat plate, are the Von Mises Boundary layer equations. Unfortunately the Von Mises transformation introduces a singularity at the plate and this discouraged the use of the equations as a means for obtaining numerical solutions of boundary layer problems in incompressible and compressible flow. In this thesis, we show that this difficulty can be overcome and the Von Mises equations are used as a basis for a finite difference evaluation of the velocity and temperature in the boundary layer adjacent to a flat plate, particular attention being given to conditions near the plate and more especially to the separation point. In the section on compressible flow, the calculations also yield a check on certain common simplifying assumptions.