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Acoustic word embeddings are fixed-dimensional representations of variable-length speech segments. Such embeddings can form the basis for speech search, indexing and discovery systems when conventional speech recognition is not possible. In zero-resource settings where unlabelled speech is the only available resource, we need a method that gives robust embeddings on an arbitrary language. Here we explore multilingual transfer: we train a single supervised embedding model on labelled data from multiple well-resourced languages and then apply it to unseen zero-resource languages. We consider three multilingual recurrent neural network (RNN) models: a classifier trained on the joint vocabularies of all training languages; a Siamese RNN trained to discriminate between same and different words from multiple languages; and a correspondence autoencoder (CAE) RNN trained to reconstruct word pairs. In a word discrimination task on six target languages, all of these models outperform state-of-the-art unsupervised models trained on the zero-resource languages themselves, giving relative improvements of more than 30% in average precision. When using only a few training languages, the multilingual CAE performs better, but with more training languages the other multilingual models perform similarly. Using more training languages is generally beneficial, but improvements are marginal on some languages. We present probing experiments which show that the CAE encodes more phonetic, word duration, language identity and speaker information than the other multilingual models.