The tremendous success of generative models in recent years raises the question whether they can also be used to perform classification. Generative models have been used as adversarially robust classifiers on simple datasets such as MNIST, but this robustness has not been observed on more complex datasets like CIFAR-10. Additionally, on natural image datasets, previous results have suggested a trade-off between the likelihood of the data and classification accuracy. In this work, we investigate score-based generative models as classifiers for natural images. We show that these models not only obtain competitive likelihood values but simultaneously achieve state-of-the-art classification accuracy for generative classifiers on CIFAR-10. Nevertheless, we find that these models are only slightly, if at all, more robust than discriminative baseline models on out-of-distribution tasks based on common image corruptions. Similarly and contrary to prior results, we find that score-based are prone to worst-case distribution shifts in the form of adversarial perturbations. Our work highlights that score-based generative models are closing the gap in classification accuracy compared to standard discriminative models. While they do not yet deliver on the promise of adversarial and out-of-domain robustness, they provide a different approach to classification that warrants further research.