A topological space $X$ is called $\pi$-\emph{normal} if for any two disjoint closed subsets $A$ and $B$ of $X$ one of which is $\pi$-closed, there exist two open disjoint subsets $U$ and $V$ of $X$ such that $A\subseteq U$ and $B\subseteq V$. We will present some characterizations of $\pi$-normality and some examples to show relations between $\pi$-normality and other weaker version of normality such as mild normality, almost normality, and quasi-normality.