We present the first systematic study on spectroscopical indicators of stellar winds and mass-loss in A-type supergiants. These stars are the coolest ones for which radiatively driven wind theory seems to be applicable. However, no systematic study has been carried out on a significant sample in order to test the correctness of this theory. We have performed an extensive spectroscopic analysis from several high resolution visible and ultraviolet observations. Besides the standard methodology to do such analysis, we have developed techniques based on Fast-Fourier transforms. We have synthesized some of the most significant lines profiles using static model atmospheres (Kurucz models) and, most importantly, we have also synthesized these profiles for an expanding atmosphere in the comoving frame by using the equivalent-two-level-atom (ETLA) approach. We can divide our sample of stars into two groups, according to the spectral features formed in the wind. The most luminous stars show characteristic spectral lines of mass outflow, both in the visible and ultraviolet spectral ranges. These stars also present strong line profile variations indicating the existence of sources of instability in the wind. On the contrary, the less luminous stars only show signs of wind in the ultraviolet resonance MgII lines as a time-evolving component. However, its evolution is much faster than the variations found in the most luminous A-supergiants. The frontier between these two groups is around an absolute magnitude of -6. The spectroscopic analysis suggests that the difference between both groups is related to the extension and density of the envelope around these objects. Although the radiation pressure seems the most plausible mechanism to drive the wind in A-type supergiants, we find discrepancies between theory and observations. The main prediction of radiatively driven wind theory is the linear relation between the escape velocity and the terminal velocity of the wind, so that the terminal velocity increases with increasing escape velocity. A-type supergiants fit this relation relatively to OB stars. But within our sample this relation does not hold: we find a scatter of values in such a way that the terminal velocity of the wind decreases with increasing escape velocities. Such scatter correlates with stellar parameters as effective temperature, radius and luminosity. This behavior was previously found for O stars, but the explanation given in that case is not meaningful for A-supergiants. Here we find a correlation between such scatter and the rotational velocities that has not been found forother stars with radiative winds. An spherically symmetric radiatively driven wind may be a good first order approximation to the envelope of these stars. Nevertheless, a more realistic model must include deviations from spherical symmetry, inhomogeneities and wind streams. In such a way, the distinction of two groups in A-supergiants would be determined by the extension and density of the envelope where the inhomogeneities are propagated.