We proposed a new framework, CROWN, to certify robustness of neural networks with general activation functions including but not limited to ReLU, tanh, sigmoid, arctan, etc. CROWN is efficient and can deliver lower bounds of minimum adversarial distortions with guarantees (the so-called certified lower bound or certified robustness). We compare CROWN with various certified lower bounds methods including Global Lipschitz constant and Fast-Lin and show that CROWN can certify much large lower bound than the Global Lipschitz constant based approach while improve the quality (up to 28%) of robustness lower bound on ReLU networks of state-of-the-art robustness certification algorithms Fast-Lin. We also compare CROWN with robustness score estimate CLEVER and adversarial attack methods (CW,EAD). Please See Section 4 and Appendix E of our paper for more details.