This paper analyzes the maximum power that a kite, or system of kites, can extract from the wind. Firstly, a number of existing results on kite system efficiency are reviewed. The results that are generally applicable require significant simplifying assumptions, usually neglecting the effects of inertia and gravity. On the other hand, the more precise analyses are usually only applicable to a particular type of kite-power system. Secondly, a novel result is derived that relates the maximum power output of a kite system to the angle of the average aerodynamic force produced by the system. This result essentially requires no limiting assumptions, and as such it is generally applicable. As it considers average forces that must be balanced, inertial forces are implicitly accounted for. In order to derive practically useful results, the maximum power output is expressed in terms of the system overall strength-to-weight ratio, the tether angle and the tether drag through an efficiency factor. The result is a simple analytic expression that can be used to calculate the maximum power-producing potential for a system of wings, flying either dynamically or statically, supported by a tether. As an example, the analysis is applied to two systems currently under development, namely, pumping-cycle generators and jet-stream wind power.