Projective planes with a large quasi-regular collineation group

London Mathematical Society Lecture Note Series n. 307 Let be a finite projective plane of order n, and let G be a large abelian (or, more generally, quasiregular) collineation group of ; to be specific, we assume |G| > (n2 + n + 1)/2. Such planes have been classified into eight cases by Dembowski and Piper in 1967. We survey the present state of knowledge about the existence and structure of such planes. We also discuss some geometric applications, in particular to the construction of arcs and ovals. Technically, a recurrent theme will be the amazing strength of the approach using various types of difference sets and the machinery of integral group rings.