  We apply the recently introduced method of hermitization to study in the
large $N$ limit non-hermitean random matrices that are drawn from a large class
of circularly symmetric non-Gaussian probability distributions, thus extending
the recent Gaussian non-hermitean literature. We develop the general formalism
for calculating the Green's function and averaged density of eigenvalues, which
may be thought of as the non-hermitean analog of the method due to Br\`ezin,
Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random
matrices. We obtain an explicit algebraic equation for the integrated density
of eigenvalues. A somewhat surprising result of that equation is that the shape
of the eigenvalue distribution in the complex plane is either a disk or an
annulus. As a concrete example, we analyze the quartic ensemble and study the
phase transition from a disk shaped eigenvalue distribution to an annular
distribution. Finally, we apply the method of hermitization to develop the
addition formalism for free non-hermitean random variables. We use this
formalism to state and prove a non-abelian non-hermitean version of the central
limit theorem.
