For L/K an extension of fields and V an algebraic variety over K say that V is Diophantine Stable for the extension L/K if V(L) = V(K). That is, if `V acquires no new rational points’ when one makes the field extension from K to L. I will describe some recent results joint with Karl Rubin regarding Diophantine Stability and give a survey of related recent statistics, heuristics, and conjectures.