We review recent results of R&D efforts directed toward increasing the operating gradients of normal conducting accelerating structures. This work includes study of the basic physics of rf breakdown phenomena and high power tests of full scale accelerating structures. The basic study includes experiments to understand the effects of rf magnetic and electric fields on statistical properties of rf breakdowns, rf breakdown dependences on materials, and experiments at cryogenic temperatures. The tests of full-scale accelerating structures show the way in which complex geometries affect rf breakdown properties. In addition to these experiments, we discuss advances in theoretical models which describe rf breakdown.