Neural Operators (NOs) have been widely applied in scientific fields such as climate science, fluid dynamics, materials science, and transportation engineering due to their ability to accurately approximate nonlinear mappings between functional spaces. However, NOs often face challenges when handling discontinuous functions, producing smeared out discontinuities or unphysical behaviors, which hinders their reliability in applications where discontinuities play a critical role. To address this, we propose the Moment Neural Operator (MNO), which leverages moment-based discretization to effectively learn mappings between discontinuous functions. This approach enhances mathematical interpretability by enforcing the learned representation to approximate moments of the target function, enabling reconstruction through established numerical methods. Experimental results show that MNO outperforms traditional NOs in approximating discontinuous functions.