  We investigate properties of a particle confined to a hard-wall spiral-shaped
region. As a case study we analyze in detail the Archimedean spiral for which
the spectrum above the continuum threshold is absolutely continuous away from
the thresholds. The subtle difference between the radial and perpendicular
width implies, however, that in contrast to `less curved' waveguides, the
discrete spectrum is empty in this case. We also discuss modifications such a
multi-arm Archimedean spirals and spiral waveguides with a central cavity; in
the latter case bound state already exist if the cavity exceeds a critical
size. For more general spiral regions the spectral nature depends on whether
they are `expanding' or `shrinking'. The most interesting situation occurs in
the asymptotically Archimedean case where the existence of bound states depends
on the direction from which the asymptotics is reached.
