  For an integer $d \geq 2$, a family $\mathcal{F}$ of sets is
$\textit{$d$-wise intersecting}$ if for any distinct sets $A_1,A_2,\dots,A_d
\in \mathcal{F}$, $A_1 \cap A_2 \cap \dots \cap A_d \neq \emptyset$, and
$\textit{non-trivial}$ if $\bigcap \mathcal{F} = \emptyset$. Hilton and Milner
conjectured that for $k \geq d \geq 2$ and large enough $n$, the extremal
non-trivial $d$-wise intersecting family of $k$-element subsets of $[n]$ is one
of the following two families: \begin{align*} &\mathcal{H}(k,d) = \{A \in
\binom{[n]}{k} : [d-1] \subset A, A \cap [d,k+1] \neq \emptyset\} \cup \{[k+1]
\setminus \{i \} : i \in [d - 1]\} \\ &\mathcal{A}(k,d) = \{ A \in
\binom{[n]}{k} : |A \cap [d+1]| \geq d \}. \end{align*} The celebrated
Hilton-Milner Theorem states that $\mathcal{H}(k,2)$ is the unique extremal
non-trivial intersecting family for $k>3$. We prove the conjecture and prove a
stability theorem, stating that any large enough non-trivial $d$-wise
intersecting family of $k$-element subsets of $[n]$ is a subfamily of
$\mathcal{A}(k,d)$ or $\mathcal{H}(k,d)$.
