The invention provides improved burners, combustion apparatus, and methods for carbon nanomaterial production. The burners of the invention provide sooting flames of fuel and oxidizing gases. The condensable products of combustion produced by the burners of this invention produce carbon nanomaterials including without limitation, soot, fullerenic soot, and fullerenes. The burners of the invention do not require premixing of the fuel and oxidizing gases and are suitable for use with low vapor pressure fuels such as those containing substantial amounts of polyaromatic hydrocarbons. The burners of the invention can operate with a hot (e.g., uncooled) burner surface and require little, if any, cooling or other forms of heat sinking. The burners of the invention comprise one or more refractory elements forming the outlet of the burner at which a flame can be established. The burners of the invention provide for improved flame stability, can be employed with a wider range of fuel/oxidizer (e.g., air) ratios and a wider range of gas velocities, and are generally more efficient than burners using water-cooled metal burner plates. The burners of the invention can also be operated to reduce the formation of undesirable soot deposits on the burner and on surfaces downstream of the burner.