We present a new method and system, called DeepZ, for certifying neural network robustness based on abstract interpretation. Compared to state-of-the-art automated verifiers for neural networks, DeepZ: (i) handles ReLU, Tanh and Sigmoid activation functions, (ii) supports feedforward and convolutional architectures, (iii) is significantly more scalable and precise, and (iv) and is sound with respect to floating point arithmetic. These benefits are due to carefully designed approximations tailored to the setting of neural networks. As an example, DeepZ achieves a verification accuracy of 97% on a large network with 88,500 hidden units under $L_{\infty}$ attack with $\epsilon = 0.1$ with an average runtime of 133 seconds.