A framework is developed for the design and analysis of single-viewpoint catadioptric cameras that use two or more mirrors. The use of multiple mirrors permits folding of the optics which leads to more compact camera designs than ones that use a single mirror. A dictionary of camera designs that use two conic mirrors is presented. We show that any folded system that uses conic mirrors has a geometrically equivalent system that uses a single conic mirror. This result makes it easy to determine the scene-to-image mapping of a conic folded system. In addition, we discuss the optical benefits of using folded systems. As an example, we choose a camera design from our dictionary and optimize its parameters via optical simulations. This design is used to construct a compact video camera that provides a hemispherical field of view.