In this paper, we provide an overview of research on multiword expressions (MWEs), from a natural language processing perspective. We examine methods developed for modelling MWEs that capture some of their linguistic properties, discussing their use for MWE discovery and for idiomaticity detection. We concentrate on their collocational and contextual preferences, along with their fixedness in terms of canonical forms and their lack of word-for-word translatatibility. We also discuss a sample of the MWE resources that have been used in intrinsic evaluation setups for these methods.