Making neural networks robust against adversarial inputs has resulted in an arms race between new defenses and attacks. The most promising defenses, adversarially robust training and verifiably robust training, have limitations that restrict their practical applications. The adversarially robust training only makes the networks robust against a subclass of attackers and we reveal such weaknesses by developing a new attack based on interval gradients. By contrast, verifiably robust training provides protection against any L-p norm-bounded attacker but incurs orders of magnitude more computational and memory overhead than adversarially robust training. We propose two novel techniques, stochastic robust approximation and dynamic mixed training, to drastically improve the efficiency of verifiably robust training without sacrificing verified robustness. We leverage two critical insights: (1) instead of over the entire training set, sound over-approximations over randomly subsampled training data points are sufficient for efficiently guiding the robust training process; and (2) We observe that the test accuracy and verifiable robustness often conflict after certain training epochs. Therefore, we use a dynamic loss function to adaptively balance them for each epoch. We designed and implemented our techniques as part of MixTrain and evaluated it on six networks trained on three popular datasets including MNIST, CIFAR, and ImageNet-200. Our evaluations show that MixTrain can achieve up to $95.2\%$ verified robust accuracy against $L_\infty$ norm-bounded attackers while taking $15$ and $3$ times less training time than state-of-the-art verifiably robust training and adversarially robust training schemes, respectively. Furthermore, MixTrain easily scales to larger networks like the one trained on ImageNet-200, significantly outperforming the existing verifiably robust training methods.