The invention relates to an arrangement of anti-torsional elements located on running gears of vehicles with a tilting body, which travel on railroad tracks. The inventive anti-torsional element is fixed, by means of one fastening joint thereof, to the side of the running gear facing the center of the body relative to the longitudinal direction, while being fixed, by means of the other fastening joint thereof, to the side of the coach body facing the closest end of the coach body. The fastening joint that is fixed to the coach body is placed higher than or at the same level as the fastening joint which is fixed to the running gear at all times, even at the maximum inclination of the coach body such that a rotational movement of the running gear extends or compresses the anti-torsional element depending on the direction of the curve when the vehicle travels on a curve while a simultaneous inclination of the coach body towards the interior of the curve compresses or extends the anti-torsional element depending on the direction of the curve, resulting in the anti-torsional element not being extended or compressed, or if so, only to a minimal extent. The anti-torsional element is thus permanently operated at the optimal working point thereof.