The epsilon factor is an invariant of a complex Galois representation that arises in the functional equation of its L-function. The magnitude of the epsilon factor, the conductor, is relatively easy to compute, but its sign, the root number, is more subtle and carries deep arithmetic information. This talk will get to the bottom of several situations in which root numbers are known or expected to express such information: the classical theory of Gaussand Kummer sums; the rank of elliptic curves; central characters of representations of reductive groups; and branching problems for such groups. Speaker(s): David Schwein (University of Cambridge)