The condensation of excitons, bound electron-hole pairs in a solid, into a coherent collective electronic state was predicted more than 50 years ago. Perhaps surprisingly, the phenomenon was first observed in a system consisting of two closely spaced parallel two-dimensional electron gases in a semiconductor double quantum well. At an appropriate high magnetic field and low temperature, the bilayer electron system condenses into a state resembling a superconductor, only with the Cooper pairs replaced by excitons consisting of electrons in one layer bound to holes in the other. In spite of being charge neutral, the transport of excitons within the condensate gives rise to several spectacular electrical effects. This article describes these phenomena and examines how they inform our understanding of this unique phase of quantum electronic matter.