  An ellipsoid is the image of a ball under an affine transformation. If this
affine transformation is over the complex numbers, we refer to it as a complex
ellipsoid. Characterizations of real ellipsoids have received much attention
over the years however, characterizations of complex ellipsoids have been
studied very little. This paper is a review of what is known about complex
ellipsoids from the point of view of convex geometry. In particular, the proof
of the Complex Banach Conjecture.
