Based on the vectorial Rayleigh-Sommerfeld diffraction formulation, a solution of the electric-magnetic wave equation is found, which represents vectorial nonparaxial off-axis Gaussian-beams whose propagation equation in free space is expressed in a closed form. The on-axis and far-field expressions of vectorial nonparaxial off-axis Gaussian beams, the propagation equation of vectorial nonparaxial Gaussian beams and the paraxial results are treated as special cases of our general expression. It is shown that the f parameter plays an impotant role in determining the beam nonparaxiality, whereas the off-axis parameters additionally affect the nonparaxial behavior of vectorial nonparaxial off-axis Gaussian beams. Moreover, unlike the on-axis case, there exists the longitudinal component of the field in the y direction for the off-axis case.