  Let X be a subvariety of $P^n$ defined by equations of degrees $ d
=(d_1,...,d_s)$, over an algebraically closed field k of any characteristic. We
study properties of the Fano scheme $F_r(X)$ that parametrizes linear subspaces
of dimension r contained in X. We prove that $F_r(X)$ is connected and smooth
of the expected dimension for n big enough (this was previously known in
characteristic 0 or for r=1). Using Bott's theorem, we prove a vanishing
theorem for certain bundles on the Grassmannian and use it to calculate the
cohomology groups of $F_r(X)$ in degree $\le \dim X-2r$, and to prove that
$F_r(X)$ is projectively normal in the Grassmannian. Finally, we prove that for
n big enough, the rational Chow group $A_1(F_r(X))$ is of rank 1, and $F_r(X)$
is unirational. All bounds on n are effective.
