I will discuss various aspects of the model theoretic approach to Manin-Mumford and functional Mordell-Lang, originating in Hrushovski's work in the 1990's. The underlying structural results were dichotomy theorems for minimal sets in difference closed fields and differentially closed fields. I will also touch on later developments: the canonical base property, the possibility of reducing function field Mordell-Lang to function field Manin-Mumford, and possibly the recent work of Roessler and Corpet.