  A Fano problem is an enumerative problem of counting $r$-dimensional linear
subspaces on a complete intersection in $\mathbb{P}^n$ over a field of
arbitrary characteristic, whenever the corresponding Fano scheme is finite. A
classical example is enumerating lines on a cubic surface. We study the
monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$
varies. We prove that the monodromy group is either symmetric or alternating in
most cases. In the exceptional cases, the monodromy group is one of the Weyl
groups $W(E_6)$ or $W(D_k)$.
