Recently, new FeAs based high-temperature superconductors CaAFe 4 As 4 (A=K, Rb, Cs) with a layered tetragonal crystal structure were synthesized (T C ~ 30 K). In this Letter, we report for the first time the band structures, Fermi surface topology, total and partial densities of electronic states and interatomic interactions for CaAFe 4 As 4 as estimated by means of the first-principles FLAPW-GGA calculations. The interatomic bonding picture can be represented as a highly anisotropic mixture of metallic, covalent, and ionic contributions, which are realized inside Fe 4 As 4 layered blocks and between these blocks and Ca, A atomic sheets. The Fermi surfaces of these systems have a multisheet character and are compiled of a large number of cylinders at the edges and in the central part of the Brillouin zone. It is established that the high-temperature superconductivity in CaAFe 4 As 4 compounds as in other related systems correlates well with such structure parameters as bond angles and anion height.