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We propose a novel decoding approach for neural machine translation (NMT) based on continuous optimisation. We reformulate decoding, a discrete optimization problem, into a continuous problem, such that optimization can make use of efficient gradient-based techniques. Our powerful decoding framework allows for more accurate decoding for standard neural machine translation models, as well as enabling decoding in intractable models such as intersection of several different NMT models. Our empirical results show that our decoding framework is effective, and can leads to substantial improvements in translations, especially in situations where greedy search and beam search are not feasible. Finally, we show how the technique is highly competitive with, and complementary to, reranking.