Group Rings and Class Groups I Some general facts II Some notes on representation theory III The leading coefficient of units IV Class sum correspondence V More on the class sum correspondence VI Subgroup rigidity VII Global units VIII Locally isomorphic group rings IX Zassenhaus conjecture X Variations of the Zassenhaus conjecture XI Group Extensions XII Class sums of p elements XIII Clifford theory revisited XIV Examples I Introduction and Review of the Tame Case II Hopf Orders III Principal Hornogeneous Spaces IV Arithmetic Applications: The Cyclotomic Case V Arithmetic Applications: The Elliptic Case References.