In this paper, we develop a methodology to analyze systematically the capacity of airborne MIMO wireless communication systems with linear transmit and receive antenna arrays. Our methodology is based on a general three-dimensional airborne MIMO communication model that we develop to accommodate arbitrary alignments of the transmit and receive antenna arrays. Specifically, we determine an upper bound and a lower bound for the airborne MIMO capacity. Then, we derive a necessary and sufficient condition for airborne MIMO communication systems to achieve the capacity upper bound. The necessary and sufficient condition allows us to properly select the system parameters and design airborne MIMO communication systems that reach the best possible performance in terms of system capacity. We prove that when the distance between the transmit and receive antenna arrays is within a certain range, there exists a set of system parameter values for which the capacity of the MIMO communication system achieves the theoretical upper bound and this capacity value is larger than the average capacity of the corresponding conventional MIMO communication system with Rayleigh fading. We also show that asymptotically, as the distance between the transmit and receive antenna arrays goes to infinity, the capacity of the airborne MIMO system converges to the capacity lower bound. Extensive numerical studies included in this paper illustrate and validate our theoretical developments.