Ordered states of cellular flames on circular porous plug burners consist of concentric rings of cells. At certain values of the flow rate and equivalence ratio a transition is made to a state in which entire rings of ceils rotate. The direction of rotation depends only on the initial conditions. Our observations of rotating cellular flames include a single rotating ring, an outer ring rotating about a single inner cell, a rotating inner ring surrounded by a fixed outer ring, and two corotating or counterrotaling concentric rings. A rotating ring of cells can also make a transition to a modulated rotating state in which the shapes of the cells and the speed of rotation periodically change. In another rotating mode, a single central cell takes the shape of a spiral which rotates inside a fixed outer ring of cells. The physical characteristics of these modes are described and comparisons with relevant theoretical studies are made.