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 → German translation task, order does not have a substantial impact on output quality, with unintuitive orderings such as alphabetical and shortest-first matching the performance of a standard Transformer. This demonstrates that traditional left-to-right generation is not strictly necessary to achieve high performance. On the other hand, results on the WMT'18 English → Chinese task tend to vary more widely, suggesting that translation for less well-aligned language pairs may be more sensitive to generation order.