In the last decade, deep artificial neural networks have achieved astounding performance in many natural language processing tasks. Given the high productivity of language, these models must possess effective generalization abilities. It is widely assumed that humans handle linguistic productivity by means of algebraic compositional rules: Are deep networks similarly compositional? After reviewing the main innovations characterizing current deep language processing networks, I discuss a set of studies suggesting that deep networks are capable of subtle grammar-dependent generalizations, but also that they do not rely on systematic compositional rules. I argue that the intriguing behaviour of these devices (still awaiting a full understanding) should be of interest to linguists and cognitive scientists, as it offers a new perspective on possible computational strategies to deal with linguistic productivity beyond rule-based compositionality, and it might lead to new insights into the less systematic generalization patterns that also appear in natural language.