The panel method is reviewed, in which linear polynomials are used to approximate the vorticity distribution on the surface of a body. Two new panel methods are developed, in which quadratic and cubic polynomials are used in an attempt to make the derivative of the velocity continuous along the surface of the body. But the results are not better than those obtained by choosing the linear polynomials as the interpolations. After the three interpolation schemes are critically evaluated, numerical examples, based on the linear scheme, are presented to illustrate some applications of the method. The flow around an airfoil in a wind tunnel is calculated, the separation of the laminar boundary layer is determined by solving the boundary-layer equations with a finite-difference scheme, and the stability of the boundary layer is investigated. It is found that the walls of the wind tunnel do not affect the separation and stability of the boundary layer significantly. Finally, the interaction of a fee vortex core with an airfoil near the ground is modeled. When the free vortex core passes the airfoil along a lower trajectory the airfoil experiences a very large thrust and suction. When along a upper trajectory, the lift and drag vary in a much smaller range. No significant effect of the ground on the airfoil is observed.