  The set of all cancellable elements of the lattice of semigroup varieties has
recently been shown to be countably infinite. But the description of all
cancellable elements of the lattice $\mathbb{MON}$ of monoid varieties remains
unknown. This problem is addressed in the present article. The first example of
a monoid variety with modular but non-distributive subvariety lattice is first
exhibited. Then a necessary condition of the modularity of an element in
$\mathbb{MON}$ is established. These results play a crucial role in the
complete description of all cancellable elements of the lattice $\mathbb{MON}$.
It turns out that there are precisely five such elements.
