This paper reports on the use of pulsed–field gradient (PFG) magnetic resonance (MR) techniques to obtain displacement and velocity spectra of steady–state, saturated flow through a column packed with glass beads. The displacement spectra obtained by PFG MR correspond to travel–distance probability–density functions (PDF) for initial conditions of a concentration impulse in a column with zero concentration. These spectra show strong dispersion–time dependence, and depart from Gaussian–shaped PDFs for short dispersion times. These data provide estimates of the dispersion–time dependence of transverse and longitudinal dispersion coefficients. The longitudinal dispersion coefficient reaches its long–time behaviour more slowly than the transverse coefficient; long–time values obtained from MR data agree well with those calculated using existing empirical correlations. A model based on three components of apparent velocities and dispersion coefficients is sufficient to describe the time dependence of displacement spectra for water flow through the bead column. The short–distance component arise because of convection–dispersion–diffusion processes within the narrow necks between particles. The long–distance component, on the other hand, represents a macroscopic convection–dispersion process. This study shows that PFG MR flow spectroscopy is a simple but potentially useful method for the investigation of flow and hydrodynamic dispersion in porous media, especially for time–dependent phenomena.