  Let $f$ be analytic in the unit disk $\mathbb D$ and normalized so that
$f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we give upper bounds of the
Hankel determinant of second order for the classes of starlike functions of
order $\alpha$, Ozaki close-to-convex functions and two other classes of
analytic functions. Some of the estimates are sharp.
