Nicolas Monod has in a recent paper introduced a new class of groups, groups with fixed-point property for cones, characterized by always admitting a non-trivial fixed-point whenever they act on cones (under some additional hypothesis). He showed that this class contains all groups of sub-exponential growth and is contained in the class of supramenable groups. (It is not known if these three classes are distinct!) He proved a number of equivalent conditions to be a group with the fixed-point property for cones, and he established a list of permanence properties for this class of groups. Monod’s results have relevance for the existence of invariant traces on a (non-unital) C*-algebra with an action of a group. The purpose of my talk will be to explain some of Monod’s results and some of their applications to C*-algebras.