Title: Limits of Algorithmic Stability for Distributional Generalization. Abstract: As machine learning models become widely considered in safety critical settings, it is important to understand when models may fail after deployment. One cause of model failure is distribution shift, where the training and test data distributions differ. In this paper we investigate the benefits of training models using methods which are algorithmically stable towards improving model robustness, motivated by recent theoretical developments which show a connection between the two.  We use techniques from differentially private stochastic gradient descent (DP-SGD) to control the level of algorithmic stability during training. We compare the performance of algorithmically stable training procedures to stochastic gradient descent (SGD) across a variety of possible distribution shifts - specifically covariate, label, and subpopulation shifts. We find that models trained with algorithmically stable procedures result in models with consistently lower generalization gap across various types of shifts and shift severities. as well as a higher absolute test performance in label shift. Finally, we demonstrate that there is there is a tradeoff between distributional robustness, stability, and performance.