One of the most intriguing phenomena in human languages is the creation and use of idiomatic expressions which defy all rules of logical composition. For instance, in Brazilian Portuguese, one can express disagreement with "nem aqui nem na China" (lit. "and-not here and-not in-the China" -> "absolutely not") or "nem que a vaca tussa" (lit. "and-not if the cow coughs" -> "absolutely never"). Idioms like these are prototypical multiword expressions (MWEs), that is, odd interpretations associated to particular word combinations.Much ink has been spilled on computational processing of MWEs since the famous ``pain-in-the-neck'' paper by Sag et al. (2002). This manuscript overviews research on this topic, with a particular focus on my own scienti...