China: Despite the destruction which the Mongols brought with them, the Chinese have tended to a perhaps surprisingly favourable view of them. The Mongols were regarded as a legitimate dynasty which possessed the Mandate of Heaven (even if many servants of the old Sung Dynasty went into retirement rather than work for them.) And they have always been given credit for expanding the frontiers of China (in Yunnan), and for restoring political unity to China - a unity it has never since lost, at least in theory. 

Central Asia: Although the "national" identities of the ex-Soviet republics of Central Asia were largely and artificial Soviet invention, it has been argued (by Peter Golden) that the modern appearance of the various Turkic groupings of Central Asia is to a great extent a consequence of Mongol rule in the area and the turbulent events (e.g. the military reorganization and redistribution of older tribal groups) which occurred at that time. Now, with the collapse of the USSR and the need for national identity and legitimacy in the new states of Central Asia, Mongol and quasi-Mongol figures such as Tamerlane have become important symbols, rather than, as previously, mere "feudal oppressors". 

Iran: It has been argued (by Bert Fragner) that modern Iran (a term not much used, until the Mongol period, since ancient times) is in some sense a creation of the Mongols. Mongol Iran had much the same borders as the modern state; the Persian language at last became supreme over Arabic; the ethnic composition of the population became much what it has since remained. All this serves to emphasize Iran's medieval orientation, which was in many ways more towards Central Asia than toward the Arab Middle East (this was much less true in more recent times). 

The rest of the Middle East: The Mongols were probably involved in the origins of the Emirate of Osman, the nucleus of the later Ottoman Empire. Iraq, once the centre of the Islamic world, became and remained a neglected frontier province. The Mongols forcibly ended the "official" Abbasid caliphate, and by doing so may well have given a boost to the Sufi (mystical) tendency in Islam. They also (inadvertently) gave a boost to the importance and centrality of Egypt, home of the Mamluk regime, the only Middle Eastern Islamic state to defy them successfully. 

Russia: The Russians tended, at the time, to go into denial about their conquest by the Mongols (Charles Halperin calls this the "Ideology of Silence"). Subsequently, they blamed the Mongols for everything that has ever gone wrong in Russia (and this is a very long list): the brutality of the political system, slowness of state formation, etc. etc - it is always convenient to have a foreign scapegoat available. Other legacies are the rise of Moscow, not an important centre before the arrival of the Mongols, but given prominence because of its ruler's role as a Mongol tax collector; and the centrality of the Orthodox Church as a pre-eminent symbol of Russian identity. 

Europe: Archibald Lewis suggested that Europe's escape from Mongol conquest and the resultant devastation was the real explanation of that much-discussed phenomenon, the "Rise of the West". This is somewhat overstated, to say the least. The main impact was probably on Europe's knowledge of the world, which was greatly expanded by the information brought back by ambassadors and merchants such as Marco Polo. When Columbus sailed in 1492, he was heading not for America but for Cathay, the land of the Great Khan; and his personal copy of Marco Polo's book still survives. The most notable possible legacy of the Mongols to Europe was a rather unfortunate one. William McNeill suggests that the Black Death, which devastated Europe in the late 1340s, may well have been carried from China to Europe along the Mongol Empire's trade routes.