Abstract: Hallucinations and off-target translation remain unsolved problems in MT, especially for low-resource languages and massively multilingual models. In this paper, we introduce two related methods to mitigate these failure cases with a modified decoding objective, without either requiring retraining or external models. In source-contrastive decoding, we search for a translation that is probable given the correct input, but improbable given a random input segment. In language-contrastive decoding, we search for a translation that is probable, but improbable given the wrong language indicator token. Experiments on the massively multilingual models M2M-100 (418M) and SMaLL-100 show that these methods suppress hallucinations and off-target translations, reducing the number of translations with segment-level chrF2 below 10 by 67-83% on average across 57 tested translation directions. In a proof of concept on out-of-English translation, we also show that we can suppress off-target translations with large language models. We release code upon acceptance.