  We classify pointed $p^3$-dimensional Hopf algebras $H$ over any
algebraically closed field $k$ of prime characteristic $p>0$. In particular, we
focus on the cases when the group $G(H)$ of group-like elements is of order $p$
or $p^2$, that is, when $H$ is pointed but is not connected nor a group
algebra. This work provides many new examples of (parametrized) non-commutative
and non-cocommutative finite-dimensional Hopf algebras in positive
characteristic.
