  Let $\mathcal{S}$ denote the class of analytic and univalent functions in
$\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ of the form $f(z)=
z+\sum_{n=2}^{\infty}a_n z^n$. In this paper, we determine sharp estimates for
the Toeplitz determinants whose elements are the Taylor coefficients of
functions in $\mathcal{S}$ and its certain subclasses. We also discuss similar
problems for typically real functions.
