Detecting out-of-distribution (OOD) inputs is critical for safely deploying deep learning models in the real world. Existing approaches for detecting OOD examples work well when evaluated on benign in-distribution and OOD samples. However, in this paper, we show that existing detection mechanisms can be extremely brittle when evaluating on in-distribution and OOD inputs with minimal adversarial perturbations which don't change their semantics. Formally, we extensively study the problem of Robust Out-of-Distribution Detection on common OOD detection approaches, and show that state-of-the-art OOD detectors can be easily fooled by adding small perturbations to the in-distribution and OOD inputs. To counteract these threats, we propose an effective algorithm called ALOE, which performs robust training by exposing the model to both adversarially crafted inlier and outlier examples. Our method can be flexibly combined with, and render existing methods robust. On common benchmark datasets, we show that ALOE substantially improves the robustness of state-of-the-art OOD detection, with 58.4% AUROC improvement on CIFAR-10 and 46.59% improvement on CIFAR-100.