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Computer technology has brought about a revolution in radiology. By combining the computer with X rays it is possible (in principle) to obtain tomographic images of any cross section in the human body. These techniques are now used for medical diagnosis in all the major hospitals of the world. But X rays are not the only kind of radiation for which computer-assisted tomography is feasible. Microwaves, electron beams, ultrasound, fast subatomic particles from accelerators, gamma rays from such sources as positron annihilation, and even magnetic fields can also be used. This paper is mainly concerned with ultrasound. Acoustic energy can often give a view of a cross section not available with X rays or other types of radiation. A mapping of acoustic and elastic discontinuities can be expected to give a basically different pattern than a mapping of X-ray absorption and scattering coefficients. Several methods of ultrasonic tomography are discussed including methods based on geometric optics and a Doppler-oriented approach. A major portion of the paper is concerned with introducing ways to take into account diffraction effects. Because of the wavelength differences, these effects are far more important for ultrasound than for X rays.