Making three-dimensional content creation easier is one of the main challenges in computer graphics. We have developed 3D modeling tools that maintain internal representations of the space of shapes that can be produced. This is enabled by representations for complex object categories, such as furniture, vehicles, and buildings. Can the space of all chairs, airplanes, or single-family homes be characterized computationally? Can generative models be designed that will synthesize custom chairs, airplanes, and houses according to specifications? Our work answers these questions affirmatively. Our representations and algorithms enable optimizing a variety of criteria over complex object classes and synthesizing novel instances from these classes.