Medieval Europeans knew that China existed…well, they knew there was a place out there that silk came from, and they called it “Seres” or “Serica”, both meaning “silk”. There are records of several groups attempting to make contact between the two empires.
It might be said fairly that the Chinese had better information about the Romans than the converse, although their information was still derived through intermediaries. It must be noted that being the state that lies between Constantinople and China is a valuable role, and that those who had it took pains to ensure that ambassadors going from one to the other were prevented for achieving success.A Chinese army reached the Caspian Sea in 97 AD, and one of its officers was sent as an ambassador to Daqin (Rome), but turned back at the shore of the Black Sea, due to stories that he had been told about the length of the journey. Excerpts from his report can be found here: http://en.wikipedia.org/wiki/Gan_Ying The son of the general also wrote a book called “The Records of the Western Regions”.
The Chinese general who had marched to the Caspian set up a series of fortresses which remained in Chinese hands for some time. The armies of Trajan, in their westernmost march into Parthia, came within a day of these garrisons. There are also stories, which lack archaeological evidence that is broadly accepted, that Romans were taken prisoner by Parthians at the Battle of Caharre, and were shipped east as warrior-slaves to man the Chinese border.
These men are said to have founded a city, called Lijen (“Alexandria”). Note that this is not the same as any of the the cities of the same name founded by Alexander. Alexander did create a city Alexandria Eschate “Alexandria the Furthest”, in Ferghana. The descendants of Alexander’s army are found all over northern India, in ancient times, and some even seem to have settled as far away as Sri Lanka.
A more modern Romanisation is Liqian. In Homer H Dubs’s History of the Former Han Dynasty we find this story: “Between 110 and 100 B.C., there arrived at the Chinese capital an embassy from the King of Parthia. Among the presents to the Chinese Emperor are stated to have been fine jugglers from Li-jien. The jugglers and dancers, male and female, from Alexandria in Egypt were famous and were exported to foreign countries. Since the King of Parthia obviously esteemed highly the Emperor of China, he naturally sent the best jugglers he could secure. When these persons were asked whence they came, they of course replied “from Alexandria,” which word the Chinese who disliked polysyllables and initial vowels and could not pronounce certain Greek sounds, shortened into “Li-jien.”. When they also learned that this place was different from Parthia, the Chinese naturally used its name for the country of these jugglers. No Chinese had been to the Roman empire, so they had no reason to distinguish a prominent place in it from the country itself. The Romans moreover had no name for their empire other than orbis terrarum, i.e., “the world,” so that these jugglers would have found it difficult to explain the name of the Roman empire! In such a fashion there probably arose the Chinese name Li-jien which, for them, denoted the Roman empire in general.” Dubs (1957) Sadly modern genetic testing and archaeology discounts this story.
In 166, a group claiming to be ambassadors from Rome appears in the Chinese court. They came by sea, from the south, which is technically possible because at this time Rome had Egyptian and Red Sea ports which linked to the Indian trade network, which linked to the Chinese one. The goods they offered the Emperor seemed southeast Asian, so there is some question as to their credentials. The Liangshu notes: “During the 5th year of the Huangwu period of the reign of Sun Quan [= CE 226] a merchant of Da Qin, whose name was Qin Lun came to Jiaozhi [Tongking]; the prefect [taishou] of Jiaozhi, Wu Miao, sent him to Sun Quan [the Wu emperor], who asked him for a report on his native country and its people. Qinlun prepared a statement and replied. At the time Zhuke [nephew to Zhuke Liang, alias Kun Ming] chastised Dan Yang [= Jiang Nan] and they had caught blackish coloured dwarfs. When Qin Lun saw them he said that in Da Qin these men were rarely seen. Sun Quan then sent male and female dwarfs, ten of each, in charge of an officer, Liu Xian of Huiji [a district in Zhejiang], to accompany Qin Lun. Liu Xian died on the road, whereupon Qin Lun returned direct to his native country.” This is sourced from the link in the next paragraph. The author notes that Qin Lun means “Leon of Rome”.
In the Third Century, there was a book in China about the products that can be found in western countries. A translation is here http://depts.washington.edu/silkroad/texts/weilue/weilue.html There are at least two embassies that claim to be from Rome that arrive in the Imperial court in the Third Century. Rome collapses in the 5th century, and Chinese power wanes after the seventh. This creates a power vacuum, and the Silk Road declines, which in turn cuts communication ties between each side. As the Mongol Empire expands, it stabilises the Silk Road, which permits Chinese people to reach back out, toward Rome, to see what is there. They are remarkably well informed. For example, look at the Da Ming Hunyi Tu map, which is Chinese from the Thirteenth Century, or the Kandigo Map from Fourteenth Century Korea.