High-performance torque tracking is crucial for accurate control of the magnitude and timing of exoskeleton assistive torque profiles. However, state-of-the-art torque control methods, e.g., iterative learning control (ILC), applied to exoskeletons cannot achieve satisfying accuracy and convergence speed. This paper aims to design a spatial iterative learning (sIL)-based torque control strategy for exoskeletons to achieve accurate and fast torque assistance, which includes a high-level controller for torque planning, a mid-level one for reference trajectory generation, and a low-level one for trajectory tracking. Compared with ILC, our proposed sIL-based control method can estimate and compensate for spatial uncertainties (e.g., joint-angle-related uncertain dynamics of the human-exoskeleton interaction system) and spatial disturbances (e.g., joint-anglerelated disturbances caused by physical interaction with the human limb) that commonly exist in exoskeletons for highly accurate torque assistance. Furthermore, our control can ensure accurate torque tracking during unsteady-state gaits with fast convergence thanks to its spatial learning capability that enables varying iterative learning speeds to deal with varying walking speeds of users for different iterations, which is not feasible by ILC methods. Experiments showed that compared with the state-of-the-art torque control methods, our sIL-based control method significantly improved the torque tracking accuracy and shortened the convergence time for both steady-state walking and unsteady-state walking (with sudden or gradual changes in gait speeds), which demonstrates its effectiveness.