  We seek wormholes among rotating cylindrically symmetric configurations in
general relativity. Exact wormhole solutions are presented with such sources of
gravity as a massless scalar field, a cosmological constant, and a scalar field
with an exponential potential. However, none of these solutions are
asymptotically flat, which excludes the existence of wormhole entrances as
local objects in our Universe. To overcome this difficulty, we try to build
configurations with flat asymptotic regions using the cut-and-paste procedure:
on both sides of the throat, a wormhole solution is matched to a properly
chosen region of flat space-time at some surfaces $\Sigma_-$ and $\Sigma_+$. It
is shown, however, that if the source of gravity in the throat region is a
scalar field with an arbitrary potential, then one or both thin shells
appearing on $\Sigma_-$ and $\Sigma_+$ inevitably violate the null energy
condition. Thus, although rotating wormhole solutions are easily found without
exotic matter, such matter is still necessary for obtaining asymptotic
flatness.
