  We define a tropical version $\F_d(\trop X)$ of the Fano Scheme $\F_d(X)$ of
a projective variety $X\subseteq \mathbb P^n$ and prove that $\F_d(\trop X)$ is
the support of a polyhedral complex contained in $\trop \Grp(d,n)$. In general
$\trop \F_d(X)\subseteq \F_d(\trop X)$ but we construct linear spaces $L$ such
that $\trop \F_1(X)\subsetneq \F_1(\trop X)$ and show that for a toric variety
$\trop \F_d(X)=\F_d(\trop X)$.
