  For a prime number $p$, a finite $p$-group of order $p^n$ has maximal class
if it has nilpotency class $n-1$. Here we examine saturated fusion systems on
maximal class $p$-groups and, in particular, we describe all the reduFor a
prime number $p$, a finite $p$-group of order $p^n$ has maximal class if and
only if it has nilpotency class $n-1$. Here we examine saturated fusion systems
$\mathcal F$ on maximal class $p$-groups $S$ of order at least $p^4$. The
Alperin-Goldschmidt Theorem for saturated fusion systems yields that $\mathcal
F$ is entirely determined by the $\mathcal F$-automorphisms of its $\mathcal
F$-essential subgroups and of $S$ itself. If an $\mathcal F$-essential subgroup
either has order $p^2$ or is non-abelian of order $p^3$, then it is called an
$\mathcal F$-pearl. The facilitating and technical theorem in this work shows
that an $\mathcal F$-essential subgroup is either an $\mathcal F$-pearl, or one
of two explicitly determined maximal subgroups of $S$. This result is easy to
prove if $S$ is a $2$-group and can be read from the work of D'\iaz, Ruiz, and
Viruel together with that of Parker and Semeraro when $p=3$. The main
contribution is for $p \ge 5$ as in this case there is no classification of the
maximal class $p$-groups.
  The main Theorem describes all the reduced saturated fusion systems on a
maximal class $p$-group of order at least $p^4$ and follows from two more
extensive theorems. These two theorems describe all saturated fusion systems,
not restricting to the reduced ones for example, on exceptional and
non-exceptional maximal class $p$-groups respectively. As a corollary, we have
the easy to remember result that states that, if $O_p(\mathcal F)=1$, then
either $\mathcal F$ has $\mathcal F$-pearls or $S$ is isomorphic to a Sylow
$p$-subgroup of $\mathrm G_2(p)$ with $p\ge 5$ and the fusion systems are
explicitly described.
