In this work we challenge the common approach of using a one-to-one mapping ('translation') between the source and target domains in unsupervised domain adaptation (UDA). Instead, we rely on stochastic translation to capture inherent translation ambiguities. This allows us to (i) train more accurate target networks by generating multiple outputs conditioned on the same source image, leveraging both accurate translation and data augmentation for appearance variability, (ii) impute robust pseudo-labels for the target data by averaging the predictions of a source network on multiple translated versions of a single target image and (iii) train and ensemble diverse networks in the target domain by modulating the degree of stochasticity in the translations. We report improvements over strong recent baselines, leading to state-of-the-art UDA results on two challenging semantic segmentation benchmarks. Our code is available at https://github.com/elchiou/Beyond-deterministic-translation-for-UDA.