Abstract:  Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H such that C_G(F) = 1. The condition C_G(F) = 1 alone already implies that G is soluble and has Fitting height bounded in terms of F. Finer results on the structure of G can be derived by using the “additional” action of H. There are good reasons to expect many properties and parameters of G to be close to the same properties and parameters of C_G(H) (possibly, also depending on |H|). Examples of such properties and parameters include the order, rank, Fitting height, nilpotency class, and exponent. I will discuss several recent results in this direction, as well as generalizations and open problems. Results concerning bounding the order, sectional rank, and Fitting height of G are based on representation theory. Various Lie ring methods are used for bounding the nilpotency class and exponent of G.