  We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N
a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and
where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded
simply connected analytic plane domain $\Omega \subset C$ relative to an
analytic weight $\rho(z) |dz|$. In the simplest case where $\Omega$ is the unit
disk and $\rho=1$, so that $P_j(z) = z^j$, it is known that the average
distribution of zeros is the uniform measure on $S^1$. We show that for any
analytic $(\Omega, \rho)$, the zeros of random polynomials almost surely become
equidistributed relative to the equilibrium measure on $\partial\Omega$ as
$N\to\infty$. We further show that on the length scale of 1/N, the correlations
have a universal scaling limit independent of $(\Omega, \rho)$.
