Turbidity currents in the ocean are driven by suspended sediment. Yet results from surveys of the modern sea floor and turbidite outcrops indicate that they are capable of transporting as bedload and depositing particles as coarse as cobble sizes. While bedload cannot drive turbidity currents, it can strongly influence the nature of the deposits they emplace. This paper reports on the first set of experiments which focus on bedload transport of granular material by density underflows. These underflows include saline density flows, hybrid saline/turbidity currents and a pure turbidity current. The use of dissolved salt is a surrogate for suspended mud which is so fine that it does not settle out readily. Thus, all the currents can be considered to be model turbidity currents. The data cover four bed conditions: plane bed, dunes, upstream-migrating antidunes and downstream-migrating antidunes. The bedload transport relation obtained from the data is very similar to those obtained for open-channel flows and, in fact, is fitted well by an existing relation determined for open-channel flows. In the case of dunes and downstream-migrating antidunes, for which flow separation on the lee sides was observed, form drag falls in a range that is similar to that due to dunes in sand-bed rivers. This form drag can be removed from the total bed shear stress using an existing relation developed for rivers. Once this form drag is subtracted, the bedload data for these cases collapse to follow the same relation as for plane beds and upstream-migrating antidunes, for which no flow separation was observed. A relation for flow resistance developed for open-channel flows agrees well with the data when adapted to density underflows. Comparison of the data with a regime diagram for field-scale sand-bed rivers at bankfull flow and field-scale measurements of turbidity currents at Monterey Submarine Canyon, together with Shields number and densimetric Froude number similarity analyses, provide strong evidence that the experimental relations apply at field scale as well.