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This work studies the sensitivity of neural networks to weight perturbations, firstly corresponding to a newly developed threat model that perturbs the neural network parameters. We propose an efficient approach to compute a certified robustness bound of weight perturbations, within which neural networks will not make erroneous outputs as desired by the adversary. In addition, we identify a useful connection between our developed certification method and the problem of weight quantization, a popular model compression technique in deep neural networks (DNNs) and a ‘must-try’ step in the design of DNN inference engines on resource constrained computing platforms, such as mobiles, FPGA, and ASIC. Specifically, we study the problem of weight quantization – weight perturbations in the non-adversarial setting – through the lens of certificated robustness, and we demonstrate significant improvements on the generalization ability of quantized networks through our robustness-aware quantization scheme.