The effects of random field errors on the propagation of a relativistic electron beam through a wiggler magnet are analyzed both theoretically and numerically. Both helical and planar wiggler configurations are studied, with and without the effects of transverse focusing forces. Theoretical expressions are derived for the random electron motion for (i) individual realizations of field errors and for (ii) ensembles of statistically identical wigglers. These results are then confirmed through three‐dimensional particle simulations of electron‐beam transport including the effects of finite emittance. In addition to producing a random walk of the beam centroid, the field errors lead to significant variations in the parallel beam energy. Asymptotically, the variance of the parallel beam energy scales asz1/2, wherezis the axial propagation distance. Although transverse beam focusing reduces the asymptotic scaling of the rms centroid displacement fromz3/2toz1/2, transverse focusing is not effective in reducing the parallel energy variation (the variance of the parallel beam energy is only reduced by a factor of &sqrt;2). Statistically, the variance of the parallel beam energy may be interpreted as an effective parallel energy spread due to field errors. In order to maintain the wave–particle resonance in small‐signal free‐electron lasers, it is desirable for this effective energy spread to be small compared to the intrinsic efficiency. As an example, in the low‐gain regime (assuming a helical wiggler in the strong‐wiggler limit), this requirement implies that the normalized rms field error must satisfy &dgr;Bˆrms<1/(2&pgr;N3/2), whereNis the number of wiggler periods.