  We study the Brown measure of certain non-hermitian operators arising from
Voiculescu's free probability theory. Usually those operators appear as the
limit in *-moments of certain ensembles of non-hermitian random matrices, and
the Brown measure gives then a canonical candidate for the limit eigenvalue
distribution of the random matrices. A prominent class for our operators is
given by polynomials in *-free variables. Other explicit examples include
R-diagonal elements and elliptic elements, for which the Brown measure was
already known, and a new class of triangular-elliptic elements. Our method for
the calculation of the Brown measure is based on a rigorous mathematical
treatment of the hermitian reduction method, as considered in the physical
literature, combined with subordination ideas and the linearization trick.
