  Let $X_\mathcal{A}$ be the projective toric variety corresponding to a finite
set of lattice points $\mathcal{A}$. We show that irreducible components of the
Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ parametrizing $k$-dimensional linear
subspaces of $X_\mathcal{A}$ are in bijection to so-called maximal Cayley
structures for $\mathcal{A}$. We explicitly describe these irreducible
components and their intersection behaviour, characterize when
$\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$
is smooth in dimension $k$, then every component of
$\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore,
in the special case $k=\dim X_\mathcal{A}-1$, we describe the non-reduced
structure of $\mathbf{F}_k(X_\mathcal{A})$. Our main result is closely related
to concurrent work done independently by Furukawa and Ito.
