It is common knowledge that translation is an ambiguous, 1-to-n mapping process, but to date, our community has produced no empirical estimates of this ambiguity. We have developed an annotation tool that enables us to create representations that compactly encode an exponential number of correct translations for a sentence. Our findings show that naturally occurring sentences have billions of translations. Having access to such large sets of meaning-equivalent translations enables us to develop a new metric, HyTER, for translation accuracy. We show that our metric provides better estimates of machine and human translation accuracy than alternative evaluation metrics.