The aiming problem in aerial gunnery has formerly been analysed by approximate methods which allow results of sufficient accuracy for practical requirements to be obtained from a very simple mathematical treatment. There has therefore been no practical demand for any more elaborate analysis, although the development of a more exact treatment of the problem is not without interest from the academic point of view. In this paper a more complete analysis is developed in a form which is generally applicable to a broader class of aiming problems. The relevant parts of the standard external ballistic analysis for projectiles fired from guns are also included for the sake of completeness, and the earlier simplified analysis of the aiming problem is also presented to allow comparison of the results with those of the more complete analysis. Finally, a section on the gunner's view of the aiming problem is included, to provide an explanation of certain difficulties which frequently arise through neglect of the important differences between the points of view of an observer moving with the gun and an observer at rest in the air.