This paper presents a geometric finite-time inner-outer loop control strategy for slung payload transportation using a quadrotor. The underactuated nature of the quadrotor in conjunction with the cable connection makes it impossible to simultaneously and arbitrarily track the load position, cable direction and the quadrotor’s attitude. By exploiting the differential flatness of the system dynamics, it is possible to arbitrarily control the load position, with the cable direction and quadrotor attitude remaining dependent on the trajectory and controller dynamics. A two-loop feedback control strategy is proposed comprising an inner-loop control law addressing quadrotor attitude tracking within finite time, and an outer-loop controller addressing the stabilization of the cable direction and position tracking of the payload trajectory. The control laws are expressed in a coordinate-free setting and are singularity-free. We include the effects of unmodeled dynamics on the load system and design an adaptive estimator to estimate and reject such uncertainties. The performance and robustness of the feedback laws is demonstrated through numerical simulations and experimental results. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.