  We study global distribution of zeros for a wide range of ensembles of random
polynomials. Two main directions are related to almost sure limits of the zero
counting measures, and to quantitative results on the expected number of zeros
in various sets. In the simplest case of Kac polynomials, given by the linear
combinations of monomials with i.i.d. random coefficients, it is well known
that their zeros are asymptotically uniformly distributed near the unit
circumference under mild assumptions on the coefficients. We give estimates of
the expected discrepancy between the zero counting measure and the normalized
arclength on the unit circle. Similar results are established for polynomials
with random coefficients spanned by different bases, e.g., by orthogonal
polynomials. We show almost sure convergence of the zero counting measures to
the corresponding equilibrium measures for associated sets in the plane, and
quantify this convergence. Random coefficients may be dependent and need not
have identical distributions in our results.
