Title: Convolution Powers as a Fourier Transform Alternative Abstract: For many questions about convolution with measures, the relevant information shows up neatly in the Fourier transform of the measure. But sometimes, when the Fourier transform is insufficient (or inapplicable), this information can be derived instead from higher powers of the convolution operator. I will discuss this approach of Fefferman and Christ, with results in Euclidean spaces and discrete nilpotent groups. (This includes joint work with Rosenblatt and Parrish.)