The results of ab initio hyperfine-interaction parameters calculations, and of x-ray diffraction and 57Fe and 151Eu Mössbauer spectroscopy study of the new 35 K superconductor CsEuFe4As4 are reported. The superconductor crystallizes in the tetragonal space group P4/mmm with the lattice parameters [Formula: see text] Å and [Formula: see text] Å. It is demonstrated unequivocally that there is no magnetic order of the Fe magnetic moments down to 2.1 K and that the ferromagnetic order is associated with the Eu magnetic moments. The Curie temperature [Formula: see text] K determined from the temperature dependence of the hyperfine magnetic field at 151Eu nuclei is shown to be compatible with the temperature dependence of the transferred hyperfine magnetic field at 57Fe nuclei that is induced by the ferromagnetically ordered Eu sublattice. The Eu magnetic moments are shown to be perpendicular to the crystallographic c-axis. The temperature dependence of the principal component of the electric field gradient tensor, both at Fe and Eu sites, is well described by a T 3/2 power-law relation. Good agreement between the calculated and measured hyperfine-interaction parameters is observed. The Debye temperature of CsEuFe4As4 is found to be 295(3) K.