The essential theoretical foundations for today's practical computer-aided design systems were laid more than two decades ago. They have served us well, but they are now sorely in need of revision. This paper suggests some directions that this revision might take. In particular, I focus on the roles of ambiguity and discontinuity in shape interpretation, instability in rules for carrying out shape computations, and non monotonicity in critical reasoning. I suggest that the challenge before us is to build a new generation of CAD systems that respond in sophisticated ways to these issues.