  We study the asymptotic distribution of the number $Z_{N}$ of zeros of random
trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as
$N$ grows to infinity, the expected number of the zeros is asymptotic to
$\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted
by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that
$\frac{Z_{N}-\E Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In
addition, we find that the analogous result is applicable for the number of
zeros in short intervals.
