In many areas of science and engineering it is of interest to
find the shape of an object or region from indirect measurements
which can actually be distilled into moments of the underlying
shapes we seek to reconstruct. In this paper, we describe
a theoretical framework for the reconstruction of a class of
planar semi-analytic domains from their moments. A part of this
class, known as quadrature domains, can approximate,
arbitrarily closely, any bounded domain in the complex
plane, and is therefore of great practical importance. We
provide an exact reconstruction algorithm of quadrature domains.
Some numerical demonstrations of the proposed algorithms will be
presented. In addition, relations of the present theory to
computer-assisted tomography and a geophysical inverse
problem will be briefly discussed.