The question of how turbidity currents erode their beds is important for understanding how submarine canyons develop, how they maintain continuity in tectonically active margins to ensure sediment bypass, and for knowing how knickpoints (reaches of anomalously steep gradient) record tectonic information. The problem is potentially more complex than fluvial erosion, because flow vigor is also affected by the flow entraining ambient water and incorporating or depositing suspended load, which can significantly affect its excess density. However, in canyon sections where the total sedimentary mass passing through the canyon is much larger than the locally excavated mass, the solid loads of eroding currents change little during passage down-canyon. Canyon morphology can then potentially reveal how gradient and other factors affect erosion rate. Simple bed erosion models are presented herein, which are analogous to the detachment- and transport-limited erosion models of fluvial geomorphology, which predict that the channel topography should advect or diffuse (smooth out), respectively. Data sets from continental slopes off Alaska, New Jersey, Oregon, Chile, the Barbados accretionary prism, and published maps from other areas show these tendencies. Although knickpoints may arise from spatially varied resistance to erosion, some of those described here lie upstream of faults or anticlines and within uniform turbidites, implying that they can advect upstream. A forward numerical model is developed for knickpoints in the southern Barbados accretionary prism, which appear to have been created in a simple manner by the frontmost thrusts. If the erosion rules are applied continuously, the channel profiles are well represented with both advective and diffusive components. If a boundary condition of nondeposition/erosion is imposed on the base of the knickpoint slope (representing scour associated with a hydraulic jump, for example), the upstream profiles can be reproduced solely by diffusion. In these channels, the threshold stress for transport or erosion is probably small relative to stress imposed by the currents, because modeling shows that a threshold sharpens the knickpoint lip rather than rounds it. For the other, mostly smaller, knickpoints studied, however, the lip varies from sharp to rounded. This varied morphology could arise from a number of influences: effects of flow acceleration, differing threshold stress, differing sediment flux affecting flow power, or depth-varying substrate resistance to erosion. Despite the diversity of forms, upstream migrations imply that erosion can be enhanced where flow is more vigorous on steep gradients, implying that the body rather than the head of turbidity currents is responsible for erosion in those cases. Also discussed is how bed failure, quarrying, and abrasive scour lead to knickpoint evolution in submarine channels that is analogous to that in fluvial channels, but also likely differences are noted.