The structure of the group ring ZG (or more generally RG for a ring R) has been of importance in both algebra and topology. On the one hand, there are purely algebraic questions that may be asked about its structure, and on the other, through K-theory, it relates to some important topics in topology. One of the most stubborn conjectures in this area is the Unit Conjecture, which aims to describe completely the structure of units in RG, where G has no non-trivial elements of finite order. In this talk I will survey the state of affairs on some of these conjectures, and describe recent work on attacking this conjecture.