The conical X-ray imaging mirror represents the long focal length limit of the Wolter type I grazing incidence mirror, in which the curved surfaces have been replaced by simple cones. When many thin-walled cones are nested, such a mirror affords the relatively high aperture filling factor needed for telescopes well suited to broadband X-ray astronomy. A conical mirror also features a constant spatial resolution across the field of view and can be designed to have an arbitrarily high intrinsic spatial resolution. We describe the general principles involved in the design of conical optics, as revealed by ray tracing studies. We also discuss the progress we have made toward the development of an arc-minute quality conical mirror, as part of the Broad-Band X-Ray Telescope experiment, which will fly as an attached Shuttle payload in late 1988. The approach we have developed for producing extremely smooth reflecting surfaces, application of a thin layer of acrylic lacquer to a metallic substrate, has produced X-ray reflecting surfaces whose microroughness is comparable with that of the highest quality polished grazing incidence mirrors. We review the efforts being made by a number of groups to produce a conical mirror with improved spatial resolution.