New high-T{c} Fe-based superconducting compounds, AFe2As2 with A=K, Cs, K/Sr, and Cs/Sr, were synthesized. The T{c} of KFe2As2 and CsFe2As2 is 3.8 and 2.6 K, respectively, which rises with partial substitution of Sr for K and Cs and peaks at 37 K for 50%-60% Sr substitution, and the compounds enter a spin-density-wave state with increasing electron number (Sr content). The compounds represent p-type analogs of the n-doped rare-earth oxypnictide superconductors. Their electronic and structural behavior demonstrate the crucial role of the (Fe2As2) layers in the superconductivity of the Fe-based layered systems, and the special feature of having elemental A layers provides new avenues to superconductivity at higher T{c}.