A series of 1,4-polybutadienes of varying molecular weight, both monodisperse and having broad or bidisperse molecular weight distributions, were studied using dielectric relaxation. The glass transition temperature, T-g, and the T-g-normalized temperature dependence of the segmental relaxation times, tau(alpha), varied monotonically with number-average molecular, M-n. Polydispersity significantly affects neither T-g nor the shape of the loss peak; the segmental relaxation dispersion is determined solely by M-n, even when the distribution of chain lengths spans molecular weights over which T-g varies. However, there is a small but significant influence of polydispersity on the T-dependence of the relaxation times, manifested as greater fragility in samples having bimodal molecular weight distributions. Properties of the prominent Johari-Goldstein (JG) secondary relaxation in 1,4-polybuladiene were measured and found to be qualitatively in accord with predictions of the coupling model. These results underscore the link between the JG and segmental processes, consistent with the JG relaxation functioning as the precursor to structural relaxation.