Tate's theorem for the Tate cohomology (agrees with group cohomology for $r\geq1$) of finite groups states the following and is used heavily:

Let $G$ a finite group and $M$ a $G$-module and suppose that for all subgroups $H$ of $G$, $H^1_T(H,M)=0$ and $H^2_T(H,M)$ is cyclic of order $|H|$. Then for all $r$ there is an isomorphism $H^r_T(G,\mathbb{Z})\cong H^{r+2}_T(G,M)$.

See Milne's notes on 'Class field theory' or the wikipedia entry for example. This is commonly applied in the case of $G$ being a Galois group $Gal(L/K)$ and $M=L^\times$ for example.