For a multi-user multiple-input multiple-output (MU-MIMO) wireless communication system, imagining that the locations of the users are now fully controllable, what is the maximum sum-capacity, and what are the corresponding optimal user locations? While these questions are irrelevant in conventional human-centric communications with random user mobility, they become critically important for emerging applications involving ground or aerial robots. This paper addresses these fundamental questions in the context of MU-MIMO communications with an unmanned aerial vehicle (UAV) swarm acting as the users. To this end, we first derive closed-form expressions for the sum-capacity of MU-MIMO UAV swarm communications. Our results reveal that, compared to conventional MU-MIMO systems, the additional degrees of freedom provided by the coordinated mobility of the UAV swarm yields substantial capacity enhancement. Specifically, when the base station (BS) is equipped with an $M$-element uniform linear array (ULA), the full spatial multiplexing gain and beamforming gain, both equal to $M$, can be achieved simultaneously. For a BS with a uniform planar array (UPA), we show that asymptotically $\frac{πM}{4}$ users can simultaneously enjoy the full beamforming gain $M$. Furthermore, we propose a novel framework to optimize UAV swarm formation for maximizing the sum-capacity achieved by successive interference cancellation (SIC) and maximizing the sum-rate via treating interference as noise (TIN), taking into account practical considerations such as collision avoidance and swarm cohesion constraints. By exploiting the manifold structure of the array response vectors with respect to UAV directions, we develop an efficient algorithm to solve the resulting non-convex formation optimization problems. Extensive simulation results demonstrate that the proposed algorithms achieve near-optimal performance.