  We show how the combination of new "linearization" ideas in free probability
theory with the powerful "realization" machinery -- developed over the last 50
years in fields including systems engineering and automata theory -- allows
solving the problem of determining the eigenvalue distribution (or even the
Brown measure, in the non-selfadjoint case) of noncommutative rational
functions of random matrices when their size tends to infinity. Along the way
we extend evaluations of noncommutative rational expressions from matrices to
stably finite algebras, e.g. type II$_1$ von Neumann algebras, with a precise
control of the domains of the rational expressions.
  The paper provides sufficient background information, with the intention that
it should be accessible both to functional analysts and to algebraists.
