The amphitheater headwalls of some bedrock canyons have been used to infer the environmental conditions and erosion processes responsible for shaping the surfaces of Earth and Mars. Morphologic identification of process is hampered, however, because we lack basic field observations and quantitative models of erosion and sediment-transport in bedrock canyons. Herein I describe five related efforts to identify erosion and transport processes through field observations and measurements of canyons on Earth, and to develop quantitative models for some of these processes. First, I present a compilation of new observations and those of others, and conclude that, despite assertions that amphitheater form is due to groundwater-seepage erosion, the evidence to support this hypothesis is ambiguous or nonexistent for most bedrock canyons. A detailed examination of two prominent examples, the Kohala valleys in Hawai‘i and Box Canyon in Idaho, has revealed no evidence for seepage erosion. Instead, field observations and topographic analyses of the Kohala valleys suggest that they likely formed by waterfall plunge-pool erosion following large-scale slumping of Kohala Volcano. In addition, sediment transport measurements and dating indicate that Box Canyon owes its origin to large-scale flooding that occurred about 45 thousand years ago. To better quantify erosion and transport processes, a mechanistic model is presented to predict the conditions under which coarse sediment is mobilized in rivers and streams over a wide range of channel-bed slopes. Lastly, I present a model for abrasion of bedrock riverbeds by impacting particles carried in bedload and suspended load.