Similarity metric learning models the general semantic similarities and distances between objects and classes of objects (e.g. persons) in order to recognise them. Different strategies and models based on Deep Learning exist and generally consist in learning a non-linear projection into a lower dimensional vector space where the semantic similarity between instances can be easily measured with a standard distance. As opposed to supervised learning, one does not train the model to predict the class labels, and the actual labels may not even be used or not known in advance. Machine learning-based similarity metric learning approaches rather operate in a weakly supervised way. That is, the training target (loss) is defined on the relationship between several instances, i.e. similar or different pairs, triplets or tuples. This learnt distance can then be applied, for example, to two new, unseen examples of unknown classes in order to determine if they belong to the same class or if they are similar. There exist numerous applications for metric learning such as face or speaker verification, image retrieval, human activity recognition or person re-identification in images. In this chapter, an overview of the principle methods and models used for similarity metric learning with neural networks is given, describing the most common architectures, loss functions and training algorithms.