This paper deals with the quadrotor craft trajectory tracking problem subject to unknown disturbance and actuator constraints. A new adaptive sliding mode control (ASMC) with finite-time convergence characteristics is proposed to guarantee quadrotor hovering in spite of parametric uncertainties and external disturbances. Compared with conventional sliding mode controller (SMC), the proposed adaptive algorithm has been developed for the vehicle altitude and attitude control, with unknown bounded of lumped uncertainties. This approach is based on a dynamical adaptive control law to avoid the overestimation and to ensure the convergence in a finite time. In addition, to solve the actuator saturation problem an auxiliary system is used. Stability analysis is demonstrated via Lyapunov theory, exhibiting that the proposed control strategy ensures that all signals of the closed-loop system are bounded and that the tracking errors are bounded in finite time. Numerical simulations and experimental results are given to illustrate the effectiveness of the proposed method, in which a comparative study with conventional SMC found in literature has been made. Experimental validation of the control strategy is carried-out using the parrot mambo mini-UAV.