Finite-time convergence control strategies based on adaptive non-singular fast terminal sliding mode are proposed for spacecraft attitude tracking subject to actuator faults, actuator saturations, external disturbances and inertia uncertainties. The designed non-singular fast terminal sliding mode control law can converge in a finite time and avoid singularity, hence it can be used to develop a finite-time fault-tolerant attitude tracking controller that meets multiple constraints. It is demonstrated that the controller is independent of inertia uncertainties and external disturbances with adaptive parameters. The controller designed considers the actuator output torque saturation amplitude requirements and makes the spacecraft accomplish certain operations within the saturation magnitude and without the need for on-line fault estimate. The Lyapunov stability analysis shows that the controller can guarantee the fast convergence of a closed-loop system and has a good fault-tolerant performance on actuator faults and saturations under the multiple constraints on actuator faults, actuator saturations, external disturbances and inertia uncertainties. Numerical simulation verified the good performance of the controller in the attitude tracking control.