We give some background about groups with a generically n-transitive action, that is, an action for which the group has a "large" orbit on the nth cartesian power of the set. Natural examples of such permutation groups arise in the classical groups, and we will present a handful of these. Our main focus will be to indicate the current state of affairs and illustrate applications of the theory to the study of primitive groups of finite Morley rank as well as to the study of sharply 2-transitive groups (not necessarily of finite Morley rank). Our definition of generic n-transitivity will be given in the context of groups of finite Morley rank. This is a class of groups, containing the algebraic groups over algebraically closed fields, which are equipped with a rudimentary notion of dimension. The talk will require no prior knowledge of Morley rank; an intuition for the way in which dimension (and degree) behave for affine varieties will suffice.