he dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. Finally, we incorporate the effect of flexibility on the dynamics of a suspension of such filaments using a mean-filed model. A dilute suspension of rigid rods settling under gravity is itself unstable to density fluctuations as a result of hydrodynamic interactions; we show that introducing filament flexibility has opposing effects on this concentration instability. On the one hand, the flexibility-induced reorientation establishes a base state that is more prone to instability, while on the other hand reorientation reduces particle aggregation thereby leading to stabilization.