  I review aspects of work done in collaboration with A. Zee and R. Scalettar
\cite{fz1,fz2,fsz} on complex non-hermitean random matrices. I open by
explaining why the bag of tools used regularly in analyzing hermitean random
matrices cannot be applied directly to analyze non-hermitean matrices, and then
introduce the Method of Hermitization, which solves this problem. Then, for
rotationally invariant ensembles, I derive a master equation for the average
density of eigenvalues in the complex plane, in the limit of infinitely large
matrices. This is achieved by resumming all the planar diagrams which appear in
the perturbative expansion of the hermitized Green function. Remarkably, this
resummation can be carried {\em explicitly} for any rotationally invariant
ensemble. I prove that in the limit of infinitely large matrices, the shape of
the eigenvalue distribution is either a disk or an annulus. This is the
celebrated ``Single-Ring'' Theorem. Which of these shapes is realized is
determined by the parameters (coupling constants) which determine the ensemble.
By varying these parameters a phase transition may occur between the two
possible shapes. I briefly discuss the universal features of this transition.
As the analysis of this problem relies heavily on summation of planar Feynman
diagrams, I take special effort at presenting a pedagogical exposition of the
diagrammatic method, which some readers may find useful.
