Erdös-style geometry is concerned with combinatorial questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be typically presented as asking for the possible number of intersections of a given (semi-)algebraic variety with large finite grids of points. An important theorem of Elekes and Szabó indicates that certain extremal asymptotic behaviors can only occur for varieties closely connected to certain algebraic structures, such as groups or fields. It turns out that techniques from model theory are very useful in recognizing these algebraic structures, and allow to obtain higher arity and dimension generalizations of the Elekes-Szabó theorem. I will overview this active area and present some recent results from joint work with Abdul Basit, Kobi Peterzil, Sergei Starchenko, Terence Tao and Chieu-Minh Tran.