The importance of e-commerce platforms has driven forward a growing body of research work on e-commerce search. We present the first large-scale and in-depth study of query reformulations performed by users of e-commerce search; the study is based on the query logs of eBay's search engine. We analyze various factors including the distribution of different types of reformulations, changes of search result pages retrieved for the reformulations, and clicks and purchases performed upon the retrieved results. We then turn to address a novel challenge in the e-commerce search realm: predicting whether a user will reformulate her query before presenting her the search results. Using a suite of prediction features, most of which are novel to this study, we attain high prediction quality. Some of the features operate prior to retrieval time, whereas others rely on the retrieved results. While the latter are substantially more effective than the former, we show that the integration of these two types of features is of merit. We also show that high prediction quality can be obtained without considering information from the past about the user or the query she posted. Nevertheless, using these types of information can further improve prediction quality.