The glass beveling machine has a table for supporting a piece of glass to be beveled, and glass bevel grinding and polishing means in the form of wheels which may have individual driving heads mounted for adjustment relative to the table for selective angle and depth of bevel. Relative movement of the grinding and polishing means and the table in a glass beveling operation is preferably effected by reciprocating the table. For holding the glass piece in place on the table during the beveling operation retaining curb means and a vacuum plenum are carried by the table. Coolant and polishing solution drain-off is facilitated by tilted mounting of the table toward collection trough means.