In 751AD Arabian and Chinese forces met in battle at a river called Talas in Central Asia. This was to mark the end of the eastward expansion of the Islamic Empire, and the westward expansion of the Chinese Empire. Discussing it on In Our Time were Hilde de Weerdt (Leiden University), Michael Höckelmann (King's College London) and Hugh Kennedy (SOAS, University of London).
Kennedy started the discussion by setting the scene for what was happening in the Arab Empire at this time. Since the Prophet's death in 632AD his followers had conquered incredible amounts of territory very quickly, and by 751AD the Islamic Empire stretched from Spain & Morocco in the west to what is now Iran in the east. Until shortly before the time of the Battle of Talas the Umayyad Caliphate was in power, with their capital in Damascus. In 750AD the Ummayyad were replaced by the Abbasid Caliphate who ruled from Baghdad. The Abbasids' power base was further east than the Ummayyads', with particularly strong support in what is now north-east Iran. The push to expand the Empire east in both cases was not just ideological, it was also economic and born out of a desire to control the lucrative trade routes to the east.
China at this time was ruled over by the Tang dynasty. Under their rulership the Chinese Empire reached its largest extent (before the Qing era), extending up to Central Asia. The Chinese were looking to protect their lucrative land trade routes - the Silk Road. So they not only had troops on their western borders, but also formed alliances with key leaders of regions along the Silk Road. What I'm refering to as Central Asia is the region that includes modern Afghanistan, Kyrgyzstan, Tajikistan, Uzbekistan and Turkmenistan. During the time period in question this was not really an area with nations or any sort of cohesive state - instead there were several princes & chiefs who ruled over particular areas or tribes. The Chinese had alliances with the ones through which the traderoutes generally ran. The experts characterised these are not quite conquest, not quite alliance - in that the Chinese pretty much left the princes to rule as they pleased but had troops stationed nearby to make sure it was suitable.
So that's the set up for this battle - the area was a volatile one with several leaders vying for power and influence. Some of these were allied with the Chinese, some of these looked more to the Arabs. Neither the Chinese nor the Arabs really wanted to extend centralised control over the region, but did want to continue to influence it. They are drawn into conflict, because one of the Chinese allies has some sort of falling out with another chief. Said chief appeals to his Arab allies for protection, and now the Chinese and the Islamic Empires are in a situation where they need to fight each other in order to hold up their own ends of their alliances.
The battle itself took place on the river Talas in July 751, I think they said it lasted five days. The Chinese forces suffered a humiliating defeat, with much of their army captured or killed. Sources on both sides give what the experts think are likely to be highly inflated casualty rates for the Chinese. The Arab sources make their own army sound small, against a larger Chinese force of around 100,000. And the Chinese sources reverse that, saying they had 30,000 (10,000 Chinese, 20,000 mercenaries) and the Arabs had 200,000. Both would be exaggerating to suit their own propaganda needs, the experts said a few tens of thousands on each side sounds plausible. The Chinese were actually defeated in large part through treachery. The actual Chinese forces were about a third of their army, the rest was made up of mercenaries from one of the Turkic peoples. During the battle these mercenaries betrayed the Chinese who were then surrounded with the Arab army on one side and the mercenary army on the other. Arab sources imply this wasn't pre-planned.
In the aftermath of this battle many many Chinese prisoners were brought to Baghdad and other parts of the Islamic Empire. One of the Chinese sources for this battle is a man who was captured and later made his way back to China, and wrote a book about his time in Baghdad. This event is sometimes credited with bringing paper-making technology to the Islamic Empire because of these prisoners. Two of the experts (Höckelmann and de Weerdt, I think) weren't convinced by this, Kennedy (I think) was more keen on it.
The battle can be held up as a big clash between East & West, and at first glance might seem to cause the halt in expansion of both these Empires. However that doesn't appear to be the case. More important is the political situation in China. Four years after the Battle of Talas was the An Lushan rebellion, which nearly deposed the Tang dynasty - so troops were called back from the frontiers and politics in China became more inward facing for a while. The land trade routes were also less lucrative because the sea ones were becoming more important. The Islamic Empire's interest in the region lessened because it was no long as economically important. So a battle that at first seems a key moment in global politics is really just a footnote.