Title: Invariance vs Robustness of Neural Networks. Abstract: Neural networks achieve human-level accuracy on many standard datasets used in image classification. The next step is to achieve better generalization to natural (or non-adversarial) perturbations as well as known pixel-wise adversarial perturbations of inputs. Previous work has studied generalization to natural geometric transformations (e.g., rotations) as invariance, and generalization to adversarial perturbations as robustness. In this paper, we examine the interplay between invariance and robustness. We empirically study the following two cases:(a) change in adversarial robustness as we improve only the invariance using equivariant models and training augmentation, (b) change in invariance as we improve only the adversarial robustness using adversarial training. We observe that the rotation invariance of equivariant models (StdCNNs and GCNNs) improves by training augmentation with progressively larger rotations but while doing so, their adversarial robustness does not improve, or worse, it can even drop significantly on datasets such as MNIST. As a plausible explanation for this phenomenon we observe that the average perturbation distance of the test points to the decision boundary decreases as the model learns larger and larger rotations. On the other hand, we take adversarially trained LeNet and ResNet models which have good \ell_\infty adversarial robustness on MNIST and CIFAR-10, and observe that adversarially training them with progressively larger norms keeps their rotation invariance essentially unchanged. In fact, the difference between test accuracy on unrotated test data and on randomly rotated test data upto \theta , for all \theta in [0, 180], remains essentially unchanged after adversarial training . As a plausible explanation for the observed phenomenon we show empirically that the principal components of adversarial perturbations and perturbations given by small rotations are nearly orthogonal