In this pedagogical review, we discuss how electrical resistance can arise in
superconductors. Starting with the idea of the superconducting order parameter
as a condensate wave function, we introduce vortices as topological excitations
with quantized phase winding, and we show how phase slips occur when vortices
cross the sample. Superconductors exhibit non-zero electrical resistance under
circumstances where phase slips occur at a finite rate. For one-dimensional
superconductors or Josephson junctions, phase slips can occur at isolated
points in space-time. Phase slip rates may be controlled by thermal activation
over a free-energy barrier, or in some circumstances, at low temperatures, by
quantum tunneling through a barrier. We present an overview of several
phenomena involving vortices that have direct implications for the electrical
resistance of superconductors, including the Berezinskii-Kosterlitz-Thouless
transition for vortex-proliferation in thin films, and the effects of vortex
pinning in bulk type II superconductors on the non-linear resistivity of these
materials in an applied magnetic field. We discuss how quantum fluctuations can
cause phase slips and review the non-trivial role of dissipation on such
fluctuations. We present a basic picture of the superconductor-to-insulator
quantum phase transitions in films, wires, and Josephson junctions. We point
out related problems in superfluid helium films and systems of ultra-cold
trapped atoms. While our emphasis is on theoretical concepts, we also briefly
describe experimental results, and we underline some of the open questions.Comment: Chapter to appear in "Bardeen, Cooper and Schrieffer: 50 Years,"
edited by Leon N. Cooper and Dmitri Feldman, to be published by World
Scientific Pres
