  \'Etale cohomology with non-invertible coefficients has some unpleasant
properties, e.g., it is not A^1-homotopy invariant and for constructible
coefficients the expected finiteness properties do not hold. In this paper we
introduce the `tame site' which is slightly coarser than the \'etale site. Tame
cohomology coincides with \'etale cohomology for invertible coefficients but is
better behaved in the general case. The fundamental group of the tame site is
the (curve-)tame fundamental group of Wiesend and Kerz/Schmidt. The higher tame
homotopy groups hopefully have a better behaviour than the higher \'etale
homotopy groups, which vanish for affine schemes in positive characteristic by
a result of Achinger.
