Current driven motions of domain walls in ferromagnetic, cylindrical nanowires are investigated by solving the Landau-Lifshitz-Gilbert equation including the adiabatic and nonadiabatic spin torque terms. Depending on the type of domain wall (transverse or vortex) and on the nonadiabaticity parameter beta different behavior of the domain wall motion has been found. A transverse domain wall shows a linear motion accompanied by a clock- or anticlockwise precession of the wall depending on the relation between the nonadiabaticity parameter beta and the Gilbert damping alpha. For alpha=beta no rotation occurs. Further, an easy way to derive the velocity equation is presented. In the case of the vortex domain wall an unexpected chirality effect has been found. Depending on the sense of rotation either a straight motion or a reversal of the rotation followed by a straight motion can be seen. Furthermore, due to the impossibility of a Walker breakdown the averaged velocity of the domain wall v is zero for all currents with beta=0 while the motion is damped by the emission of spin waves for higher currents and beta > alpha.