The basic problem in computerized tomography is the reconstruction of a function from its line or plane integrals. Applications come from diagnostic radiology, astronomy, electron microscopy, seismology, radar, plasma physics, nuclear medicine and many other fields. More recent kinds of tomography replace the straight line model by an inverse problem for a partial differential equation.
The outline of this paper is as follows. In section 2 we survey the mathematical models used in tomography. In section 3 we give a fairly detailed survey on 2D reconstruction algorithm which still are the work horse of tomography. In section 4 we describe recent developments in 3D reconstruction. In section 5 we make a few remarks on the beginning development of algorithms for non-straight-line tomography.