  We use cluster structures and mirror symmetry to explicitly describe a
natural class of Newton-Okounkov bodies for Grassmannians. We consider the
Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well as the mirror dual
Landau-Ginzburg model $(\check{X}^\circ, W_q:\check{X}^\circ \to \mathbb C)$,
where $\check{X}^\circ$ is the complement of a particular anti-canonical
divisor in a Langlands dual Grassmannian $\check{X} = Gr_k((\mathbb C^n)^*)$,
and the superpotential W_q has a simple expression in terms of Pl\"ucker
coordinates. Grassmannians simultaneously have the structure of an
$\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a
cluster seed G, we consider two associated coordinate systems: a $\mathcal
X$-cluster chart $\Phi_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal
A$-cluster chart $\Phi_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. To
each $\mathcal X$-cluster chart $\Phi_G$ and ample `boundary divisor' $D$ in
$X\setminus X^{\circ}$, we associate a Newton-Okounkov body $\Delta_G(D)$ in
$\mathbb R^{k(n-k)}$, which is defined as the convex hull of rational points.
On the other hand using the $\mathcal A$-cluster chart $\Phi_G^{\vee}$ on the
mirror side, we obtain a set of rational polytopes, described by inequalities,
by writing the superpotential $W_q$ in the $\mathcal A$-cluster coordinates,
and then "tropicalising". Our main result is that the Newton-Okounkov bodies
$\Delta_G(D)$ and the polytopes obtained by tropicalisation coincide. As an
application, we construct degenerations of the Grassmannian to toric varieties
corresponding to these Newton-Okounkov bodies. Additionally, when $G$
corresponds to a plabic graph, we give a formula for the lattice points of the
Newton-Okounkov bodies, which has an interpretation in terms of quantum
Schubert calculus.
