This article presents a model predictive control (MPC) controller and its novel application to a hybrid tilt-quadrotor fixed-wing aircraft, which combines vertical takeoff and landing (VTOL) capabilities with high-speed forward flight. The developed MPC controller takes a velocity command from the pilot and then computes optimal attitude setpoints and propeller-tilt angles that are supplied to a fast inner attitude controller. A control allocation algorithm then maps the output of the inner attitude loop to actuator commands. The proposed MPC and control allocation of this article constitute a unified nonlinear control approach for tilt-rotor VTOL aircraft, valid in all flight modes and transitions in between. The whole approach is verified both in simulations and in real-world outdoor experiments with a remote controlled VTOL aircraft transitioning from hover to high speed and vice versa in a stable and controlled manner. Results show superior performance compared to the common binary-switch transition strategy between multicopter flight mode and the fixed-wing flight mode. The MPC controller also consistently performs better than a previously developed fused-PID control architecture in our tests.