The concept of diffraction tomography and the signal processing issues associated with it are reviewed. Two problems in signal processing are singled out and discussed in some detail. The first problem deals with the reconstruction of an image from appropriately transformed measured data that samples the spatial-frequency plane along circular arcs. A new Fourier-domain algorithm is presented for this task, and a method of extrapolating the data in the Fourier domain for missing profiles is discussed. The second problem is that of unwrapping the phase of the measured profiles. An algorithm is presented that makes use of certain physical assumptions on the object and the scattered field to accomplish this.