Arc spaces play a central place in recent developments in singularity theory and birational geometry, most notably through motivic integration. The aim of this talk is to present an overview of some recent results about the structure of arc spaces of singular varieties. Taking as a guiding thread a geometric lemma of Denef and myself going back to the early days of motivic integration, I will present some recent work of Reguera, Bouthier - Kazhdan and de Fernex - Docampo.