We review a large multidisciplinary effort to develop a family of autonomous robots capable of rapid, agile maneuvers in and around natural and artificial vertical terrains such as walls, cliffs, caves, trees and rubble. Our robot designs are inspired by (but not direct copies of) biological climbers such as cockroaches, geckos, and squirrels. We are incorporating advanced materials (e.g., synthetic gecko hairs) into these designs and fabricating them using state of the art rapid prototyping techniques (e.g., shape deposition manufacturing) that permit multiple iterations of design and testing with an effective integration path for the novel materials and components. We are developing novel motion control techniques to support dexterous climbing behaviors that are inspired by neuroethological studies of animals and descended from earlier frameworks that have proven analytically tractable and empirically sound. Our near term behavioral targets call for vertical climbing on soft (e.g., bark) or rough surfaces and for ascents on smooth, hard steep inclines (e.g., 60 degree slopes on metal or glass sheets) at one body length per second.