  Ribbons are long narrow strips possessing three distinct material length
scales (thickness, width, and length) which allow them to produce unique shapes
unobtainable by wires or filaments. For example when a ribbon has half a twist
and is bent into a circle it produces a M\"obius strip. Significant effort has
gone into determining the structural shapes of ribbons but less is know about
their behavior in viscous fluids. In this paper we determine, asymptotically,
the leading-order hydrodynamic behavior of a slender ribbon in Stokes flows.
The derivation, reminiscent of slender-body theory for filaments, assumes that
the length of the ribbon is much larger than its width, which itself is much
larger than its thickness. The final result is an integral equation for the
force density on a mathematical ruled surface, termed the ribbon plane, located
inside the ribbon. A numerical implementation of our derivation shows good
agreement with the known hydrodynamics of long flat ellipsoids, and
successfully captures the swimming behavior of artificial microscopic swimmers
recently explored experimentally. We also study the asymptotic behavior of a
ribbon bent into a helix, that of a twisted ellipsoid, and we investigate how
accurately the hydrodynamics of a ribbon can be effectively captured by that of
a slender filament. Our asymptotic results provide the fundamental framework
necessary to predict the behavior of slender ribbons at low Reynolds numbers in
a variety of biological and engineering problems.
