Abstract: Per Enflo introduced roundness as an invariant for classifying topological linear spaces up to uniform homeomorphism. In the first part of the talk we will describe how this invariant was used and extend Enflo's ideas to compare, up to uniform homeomorphism, normed and quasi-normed spaces. In the second part, we view roundness as a geometric invariant of metric spaces. It turns out that the properties of metric spaces of non-trivial roundness are very similar to the ones of non-positively curved spaces. In the last part of the talk we compute the roundness of certain Cayley graphs and connect roundness to algebraic properties of groups.