Abstract: In this work, we present an empirical study of generation order for machine translation. Building on recent advances in insertion-based modeling, we first introduce a soft order-reward framework that enables us to train models to follow arbitrary oracle generation policies. We then make use of this framework to explore a large variety of generation orders, including uninformed orders, location-based orders, frequency-based orders, content-based orders, and model-based orders. Curiously, we find that for the WMT'14 English $\to$ German and WMT'18 English $\to$ Chinese translation tasks, order does not have a substantial impact on output quality. Moreover, for English $\to$ German, we even discover that unintuitive orderings such as alphabetical and shortest-first can match the performance of a standard Transformer, suggesting that traditional left-to-right generation may not be necessary to achieve high performance.