At school, I remember being taught that the the Song dynasty succeeded the Tang dynasty in ruling China, followed by the Yuan dynasty; however, I believe that I also read from somewhere that the Mongols considered their own dynasty to be a successor state to the Jin dynasty, rather than the Song dynasty.
Note: I am referring to the Jin dynasty that predated the Mongol conquests of China
The Yuan dynasty (Mongols) actually recognized the Song, Xia, and Jin as legitimate predecessors. The Mongols had conquered the Xia, then the Jin, and, significantly later, the Southern Song (the Northern Song fell to the Jin in 1126-1127). The Liao had previously been conquered by the Jin by 1125, about 100 years before the Mongols arrived on the scene.
In medieval and early modern Chinese historiography, the perceived legitimacy of a defunct dynasty generally depended on its relation to (and recognition by) the one currently in power. The Song traced their own claim to legitimacy directly back to the Tang, via the "Five Dynasties" that controlled significant parts of northern China between the fall of the Tang and the military coup that brought the Song founder to power about 50 years later. The last Tang emperor had been deposed and replaced by a powerful warlord, Zhu Quanzhong, whose Later Liang dynasty was deposed and replaced by a succession of short-lived states (Later Tang, Later Jin, and Later Han) prior to the gradual reunification of China under the Later Zhou and then the Song. Because the Song inherited this tradition, it became their link to the Tang. And because the Song became the first dynasty since the Tang to reunify China proper, simplified versions of the dynastic cycle often list them right after the Tang, although the real historical sequence was nowhere near as simple as that.
So the Song were not considered direct successors of the Tang, but rather the inheritors of a tradition of legitimacy "passed on" directly from the Tang.