  In this article, we first briefly introduce the history of the Weil-\'etale
cohomology theory of arithmetic schemes and review some important results
established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-etale
cohomology to $S$-integers and compute the cohomology for constant sheaves
$\mathbb{Z}$ or $\mathbb{R}$. We also define a Weil-\'etale cohomology with
compact support $H_c(Y_W, -)$ for $Y=Spec \mathcal{O}_{F,S}$ where $F$ is a
number field, and computed them. We verify that these cohomology groups satisfy
the axioms state by Lichtenbaum. As an application, we derive a canonical
representation of Tate sequence from $RGamma_c(Y_W,\mathbb{Z})$. Motivated by
this result, in the final part, we define an \'etale complex $RGm$, such that
the complexes $\mathbb{Z}$-dual of the complex
$\RG(U_{et},R\Gm),\,\mathbb{Z})[2]$ is canonically quasi-isomorphic to
$\tau^{\leq 3}\RG_c(U_W,\mathbb{Z})$ for arbitrary \'etale $U$ over $Spec
\mathcal{O}_{F}$. This quasi-isomorphism provides a possible approach to define
the Weil-etale cohomology for higher dimensional arithmetic schemes, as the
Weil groups are not involved in the definition of $R\Gm$.
