We present two depth- and phase-resolving models, based on single- and two-phase approaches for suspended sediment transport under water waves. Both models are the extension of a wave hydrodynamic model Cornell Breaking Wave and Structure (COBRAS). In the two-phase approach, dilute two-phase mass and momentum equations are calculated along with a fluid turbulence closure based on balance equations for the fluid turbulence kinetic energy kf and its dissipation rate $epsilon$f. In the single-phase approach the fluid flow is described by the Reynolds-Averaged Navier-Stokes equations, while the sediment concentration is calculated by an advection-diffusion equation for the conservation of sediment mass. The fluid turbulence is calculated by kf-$epsilon$f equations that incorporate the essential influence of sediment, which can also be consistently deduced from the two-phase theory. By adopting a commonly used sediment flux boundary condition near the bed the proposed models are tested against laboratory measurements of suspended sediment under nonbreaking skewed water waves and shoaling broken waves. Although the models predict wave-averaged sediment concentrations reasonably well, the corresponding time histories of instantaneous sediment concentration are less accurate. We demonstrate that this is due to the uncertainties in the near-bed sediment boundary conditions. In addition, we show that under breaking waves the near-bed sediment pickup cannot be solely parameterized by the bottom friction, suggesting that other effects may also influence the near-bed sediment boundary conditions.