The present status of the fully-relativistic nonperturbative calculations of the fundamental atomic processes with twisted electrons is presented. In particular, the elastic (Mott) scattering, the radiative recombination, and for the very first time, the Bremsstrahlung processes are considered. The electron-ion interaction is accounted for in a nonperturbative manner, that allows obtaining reliable results for heavy systems. We investigate the influence of the "twistedness" of the incoming electron on the angular and polarization properties of the emitted electrons and photons for the elastic and inelastic scattering, respectively. It is found that these properties exhibit a strong dependence on the opening angle of the vortex electron beam in all processes considered.