This paper addresses the challenges of the design, development and control of a new convertible VTOL tailsitter unmanned aerial vehicle that combines the advantages of both fixed wing and rotary wing systems. Wind tunnel measurements are used to get an understanding of the control allocation and to model the static forces and moments acting on the system. Based on the derived model, a novel controller that operates in SO(3) and handles the dynamics of the vehicle at any attitude configuration, including the rotorcraft and fixed-wing regimes as well as their transitions, is presented. This unified controller allows the autonomous transition of the system without discontinuities of switching, as well as its overall high performance flight control. The capabilities and flying qualities of the platform and the controller are demonstrated and evaluated by means of extensive experimental studies.