The idea of paper currency in China actually long predates the Song Dynasty. A fairly natural outgrowth of the A) widespread use of block-printing techniques, and B) a downward-spiraling economic system based on hard specie, in an empire that was frequently at risk of running out of that specie, paper used as “transferrable notes of value” were put into extremely limited service as intra-governmental bills of transfer as early as the mid-Tang Dynasty, circa 800 CE. These paper notes, called “flying cash” – largely because of their propensity to blow away in the wind – shouldn’t be confused with the modern sense of paper money, as they were officially only to be used for large transfers within the government and redeemed at the capital for actual cash-coin currency. Still, it’s clear enough that the ease of transferability meant that such notes were also “unofficially” used in a limited sense, as what amounts to currency even by non-government traders within the Tang system. In any case, that would set the stage for the Song to pick up the torch for the genuine roll-out of paper money beginning in the mid-11th century.

Initially the paper currency was used only regionally, specifically in Hedong, Shanxi, and Sichuan in order to “compensate for an unmet demand for currency and to protect the Song from the outflow of copper coin from the frontier circuits to Xi Xia or Liao, both of which were deficient in metals.” Xi Xia [alt. Western Xia] and the Liao Dynasty were Song China’s primary neighbor states to its northwest and northeast, respectively. Liao in particular had long been a bogeyman for the Song, and had even outright politically and militarily dominated several of its predecessor northern states during the Five Dynasties period (907-960 CE). As recently as 1004, the Liao had been either raiding or outright invading the Song territories, which finally came to a formalized end that year through a treaty in which the Song was forced to make a number of large concessions, including annual payments (but not tributes) to the Liao in the form of silver – specifically 100,000 grams/year. Even though the Song was able to more than make this payment up and then some with the interstate trading with Liao and Xia, the outflow of bullion was a large enough concern that such bulwarks were being implemented at the local and regional levels in certain vulnerable areas as early as the 1030s. Still, it's worth noting that these were very much trial-runs, limited in terms of their regionality, and also fully backed by hard specie reserves.

Paper currency would remain a regional phenomenon for the following century, until fairly immediately following the unfavorable conclusion of the Jin-Song Wars that saw China lose its northern territories and capital to the ascendant Jurchen people that had supplanted the Khitan Liao with Chinese help, then turned on China itself, establishing the Jin Dynasty across the north. Though debilitating to morale and prestige, though, economically and militarily, Song China remained in pretty decent shape, since the southlands had long since eclipsed the north in terms of economic and cultural output. In any case, the paper system would be expanded nationally by the year 1060 under the regime of Emperor Gaozong, with the paper currency called huizi (會子). And at first it all went pretty well, but even from the outset the risk of economic instability stemming from the paper currency ran high. From Gong Wei An:

In financial matters [Emperor] Xiaozong [r. 1162-1189] stressed proper control and careful management of the state’s economy. His monetary policy was prudent and included strict control over the circulation of paper money. The main paper currency of the Southern Song, known as huizi, had first been issued by the government in 1160 during Gaozong’s reign. Its convenience was widely recognized, but, in a few years time, its unit value began to drop because of overissue. In 1167, in an attempt to remedy the situation, Xiaozong released two million taels of silver from the imperial treasury to be exchanged in the mar- kets for huizi notes that were then destroyed. This strict control of the amount of paper currency issued and the money’s popularity, especially among the merchant community, led to a shortage of notes in the Liang-Huai region (modern Jiangsu-Anhui) in 1175, and more notes had to be issued to meet the demand. In 1175, a half million strings worth of huizi was designated for circulation in the Huai region in exchange for copper coins. The emperor continued to manage paper currency with extreme care. In 1183, on seeing the number of notes that had been newly issued, Xiaozong expressed concern that the increase might lead to a drop in value. In 1185 he reiterated his policy that the paper money should not be issued in large numbers and that its circulation should be curtailed once military expenses were reduced. Xiaozong sought to check the devaluation of the huizi by guarding strictly against overissuing paper money, a policy that contributed to stability in the Southern Sung economy.

This heavy risk of inflation-though-overprinting would be the bugaboo of paper currency for virtually all of its early adopters, and is indeed pointed out as an almost textbook case of “Gresham’s Law” stating that “bad money will drive out good”. The conquering Jurchen Jin, for instance, almost immediately adopted Song-inspired paper currency, owing to their own historic dearth of hard specie such as copper. As early as 1157 – 3 years before the Southern Song even “officially” rolled out their own huizi notes, the Jin Imperial court – having only just completed their move from Huining in Manchuria to their new capital at Zhongdu (mod. Beijing).

By 1154, the Jin court had established the Bureau of Paper Currency ( at Kaifeng (the former Song capital city). The Bureau

So long as the Jin government continued to accept their bills at face value and strictly limit the number of new bills they put into their own economy (while initially retiring all bills after 7 years in rotation), the proverbial horse dung hit the silken hand-fan in 1206 when Mongol attacks against the Jin began in earnest. From Franke:

The enormous costs of the campaign severely taxed the financial resources of the state and apparently led to the excessive printing of paper money, especially when the Mongols attacked. Denominations as high as one thousand kuan were printed and put into circulation. From then on, the monetary history of Chin is a series of desperate attempts to stabilize the currency by repeated issues of paper money. Every few years a new type of money was issued with a new high-sounding name, but the value in each case sank so low that, for example, in 1221, eight hundred kuan in paper currency was equivalent to one ounce of silver. One reason for the rapid depreciation was that with every new issue the old notes continued to circulate, so that the national economy was flooded with paper money. Silver, of course, kept its value but was obviously hoarded by everybody who could find it. In the four years between 1217 and 1221, paper money had been devalued at the rate of 40,000 to 1.

This process of hyperinflation would continue up until the very end of Jin, with the last emperor, Aizong, and his court were issuing bills from their final bastion in Cai Prefecture – theoretically redeemable for silver - before being destroyed just a few months later in 1233.

The Mongols were, like the Jin and Liao before them, eager adopters of Chinese governmental and economic systems – eventually including their own form of paper money, as well. The first Mongol paper money would be rolled out under Möngke Khan, the 4tth supreme ruler of the Mongol Empire and brother of Kublai – the man who would conquer the Southern Song and found the Yuan Dynasty. Under Möngke’s direction, the Superintendency of Paper Money (Jiaochao Tizhusi) was established in 1253 and, from Allsen:

There is no doubt that the superintendency actually issued bills, for Rubruck examined specimens in Khara Khorum in 1254, but the extent to which this currency circulated and its role in the payment of k'o-ch'ai are not indicated in the Chinese sources. However, as Rubruck expressly states elsewhere in his work that the Chinese daily paid their Mongolian overlords fifteen hundred iascot (Turkic, yastuq, "pillow" or ingot) of silver in addition to an unspecified amount of silk,"8 it seems permissible to infer that paper money was not widely used for tax purposes, at least in the early years of Mongke's reign.