The free-propagation features of light beams whose transverse electric field lines are logarithmic spirals (namely, spirally polarized beams) are investigated in both the paraxial and the nonparaxial regime. The complete propagated electric field is considered, and some general properties are obtained regardless of the specific transverse distribution. Simple and significant analytical results are obtained when the transverse intensity profile is chosen as that pertinent to an axially symmetric Laguerre-Gaussian beam of order 1 (namely, spirally polarized donut beams). In particular, it is found that for such beams, the propagated longitudinal electric field can be expressed as a simple superposition of elegant Laguerre-Gaussian beams. Numerical results are presented for different values of the beam parameters and are compared with recently obtained experimental results.