Abstract: This invention provides an improved machine for finishing the edges of glass or other brittle materials. The machine is distinguished by its few moving parts, and its ease of maintenance. The edge-finishing machine includes a main conveyor section including two toothed main belts that grip and convey the glass or other material to be finished past grinding or polishing wheels, or other means for finishing the edges of the material. These main conveyor belts are synchronously driven, each by a separate motor or other power transmission unit. The main conveyor section of the machine also includes means to tension the main belts, preferably including non-rotating returns made from durable, low-friction material such as ultra-high molecular weight plastic.