The first part of the paper reviews the theory of the single-heading flight of an airplane on a plane surface with unchanging geostrophic wind and shows that the simplicity of the formula for the heading of such a flight is lost if the surface is spherical or if the wind field is changing. It is also shown that the single-heading flight is neither necessarily faster nor necessarily slower than the straight-line flight. In the second part, the problem of determining the quickest flight path between two given points for a given air speed is discussed. When the wind is not everywhere zero, time can frequently be saved by deviating from the great-circle route to take advantage of stronger tail winds or weaker headwinds. This is especially true on longer flights. To determine the paths having the least possible flight time for a given wind field (in general varying in both space and time) is a fairly complicated problem in the calculus of variations. It was first solved by Zermelo in 1930; his solution is applicable, however, only to flat surfaces. The solution has now been extended to cover the case of flight on the surface of a sphere, such as the earth. The solution takes the form of a differential equation which the airplane's heading is to satisfy. A similar equation for flight in three dimensions is discussed.