Robots have unusual potential to interactively support human design thinking. As physical and social actors, robots can be programmed to directly engage in the kind of materially-situated, social design conversations that typify human design activity. As computational agents, robots can also inject perspectives, knowledge, and approaches to complement human design thinking. In this dissertation, we make this argument from theory and prior work and describe observations from two primary and two supporting studies of what robots designing with humans might look like. The first primary study examines how a human and a robot might collaboratively design a solution to a problem, while the second examines how they might define design problems together. We reflect on insights across these two studies and propose next steps aimed at isolating the importance of social and physical interaction in this context. We conclude by defining key questions at the core of the study and pursuit of human-robot collaborative design (HRCD).