A sewing system serves for sewing seams which the multi-dimensionally in space. A needle-bar upper part of a sewing head is mounted so as to be pivotable in a driven manner about a needle-bar longitudinal axis. A gripper lower part is mounted so as to be pivotable in a driven manner, and in a manner synchronous with the needle-bar upper part, about the needle-bar longitudinal axis, in the sewing-head lower part. Motor components for driving the movement of a needle bar, for pivoting the needle-bar upper part, for driving a gripper in a manner synchronous with the needle bar and for pivoting the gripper lower part are disposed outside the needle-bar upper part and outside the gripper lower part. Parts to be sewn that are topographically non-uniform are also rendered accessible.