In this work, we investigate the possibilities of reaching wide-angle perfect absorption with Huygens' metasurfaces possessing both electric and magnetic responses. We analyze the role of the anisotropy of the polarizability tensor of the unit cell and reveal the conditions for the optimal angular stability in symmetric absorption as well as polarization independence. Then we approximate the desirable behavior with a Huygens' metasurface whose reflectance is low in a wide frequency range while the absorption is total around a single resonance frequency. Reaching of the optimal angular stability was numerically demonstrated and the corresponding wide-angle and wide-band properties were experimentally checked.