  We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$
where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and
the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly
supported measure $\tau$ satisfying the Bernstein-Markov property on a regular
compact set $K \subset \mathbb{C}$. We show that if
$\mathbb{P}(|\xi_0|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure
of the zeros of $H_n$ converges weakly in probability to the equilibrium
measure of $K.$ This is the best possible result, in the sense that the roots
of $G_n(z)=\sum_{j=0}^n\xi_jz^j$ fail to converge in probability to the
appropriate equilibrium measure when the above condition on the $\xi_j$ is not
satisfied. In addition, we give a multivariable version of this result.
  We also consider random polynomials of the form
$\sum_{k=0}^n\xi_kf_{n,k}z^k$, where the coefficients $f_{n,k}$ are complex
constants satisfying certain conditions, and the random variables $\{\xi_k\}$
satisfy $\mathbb{E} \log(1 + |\xi_0|) < \infty$. In this case, we establish
almost sure convergence of the normalized counting measure of the zeros to an
appropriate limiting measure. Again, this is the best possible result in the
same sense as above.
