This paper proposes Neural Robot Dynamics (NeRD), a generalizable neural network-based robot simulator to address the challenge of accurately and efficiently simulating modern robots with high degrees of freedom and complex mechanisms. NeRD learns a specific dynamic model for a robot composed of articulated rigid bodies and predicts future states under contact constraints. To address the inherent application-specific learning and generalization failures of existing neural network simulators, NeRD employs a robot-centric and spatially invariant simulation state representation. It integrates the low-level dynamics and contact solvers of existing analytical simulators with the NeRD model, and bridges the gap between simulation and reality through fine-tuning using real-world data. Experimental results demonstrate that the NeRD simulator is stable and accurate across thousands of simulation steps, generalizes well across task and environment configurations, and enables policy learning unique to a neural network engine.