/u/Tiako is probably best placed to provide a detailed account of earlier instances of Roman contact with China, but I have written a little bit on what the (Eastern) Romans knew of China in the sixth and seventh centuries here. Crucially, after nearly two centuries of zero official contact between Rome and China (at least according to the Chinese sources), in 643 a Roman embassy once again reached China. This would be repeated several times in the following decades, which is indicative of how the two empires were aware of each other's status and tried to communicate with each other. Unfortunately, it is impossible to piece together whether there was any unofficial contact between people from the two empires in this period, even though it was entirely possible for traders to travel very far away from home. Nor do we have any Greek sources for what the court in Constantinople thought about China, aside from the account of Theophylact I quoted. This description remains understudied and I only know of a suggestion that it was a garbled version of some information relayed by Turkish envoys. Beyond that, we can only speculate, especially as we literally have no way to prove how Theophylact got that information.

the Romans were definitely aware of China and there were Chinese travellers that made it to Rome. China created the concept of "Da Qin" (literally big empire) to describe the Romans, placing their lands on a map west of Parthia, and stated the people of Da Qin. Some quotes from the writings of Gan Ying:

The Kingdom of Da Qin (the Roman Empire) is also called Lijian. As it is found to the west of the sea, it is also called the Kingdom of Haixi ("West of the Sea"). The territory extends for several thousands of li. It has more than four hundred walled towns. There are several tens of smaller dependent kingdoms. The walls of the towns are made of stone. They have established postal relays at intervals, which are all plastered and whitewashed. There are pines and cypresses, as well as trees and plants of all kinds

Their kings are not permanent. They select and appoint the most worthy man. If there are unexpected calamities in the kingdom, such as frequent extraordinary winds or rains, he is unceremoniously rejected and replaced.

They also have a fine cloth which some people say is made from the down of 'water sheep', but which is made, in fact, from the cocoons of wild silkworms. They blend all sorts of fragrances, and by boiling the juice, make a compound perfume. [They have] all the precious and rare things that come from the various foreign kingdoms. They make gold and silver coins. Ten silver coins are worth one gold coin.

This next one is vital

The king of this country always wanted to send envoys to the Han, but Anxi, wishing to control the trade in multi-coloured Chinese silks, blocked the route to prevent [the Romans] getting through [to China].

Anxi = Parthia

The descriptions are vivid and accurate in many cases, and it's clear the Romans are aware of China and have received Chinese visitors. However, Parthia prevented any Romans from visiting China, to control the trade, or simply because Roman-Parthian relations were constantly hostile.

This question gets asked a lot. Here's an excellent older thread about it: https://www.reddit.com/r/AskHistorians/comments/wszki/to_what_extend_did_han_china_and_the_roman_empire/

Han records indicate a Roman embassy made it to China in 166. I was just at a seminar in which a scholar indicated that it was sent by Gaius Avidius Cassius, on his own initiative, in the name of Emperor Marcus Aurelius Antoninus, in pursuit of prestige and wealth, and in concert with merchant supporters in the Roman East. Unfortunately, when the envoys arrived, it was when the Han were in the midst of political turmoil, and contact with a distant empire wasn't felt to be very important.

Here's a link to the seminar, the accompanying paper hasn't been published yet: “The First Official Contact between Rome and China?: a Roman Embassy to the Han Emperor’s Court in A.D. 166”

If you mean direct, diplomatic contact, the answer is probably no. There was an incident in 166 CE that people love to bring up mentioned in the Hou Han Shu, a fifth century history, that I will just copy out:

This lasted till the ninth year of the Yen-hsi period during the emperor Huan-ti's reign [166 C.E.] when the king of Ta-ts'in, An-tun [Marcus Aurelius Antoninus], sent an embassy who, from the frontier of Jih-nan [Annam][Vietnam] offered ivory, rhinoceros horns, and tortoise shell. From that time dates the direct intercourse with this country. The list of their tribute contained no jewels whatever, which fact throws doubt on the tradition.

The goods offered by the "ambassadors" isn't just paltry, but is also exactly the sort of goods one would get from south and southeast Asia. The Roman embassies we actually do know about, such as the one to Sri Lanka, tended to have things like coral, gold, finely worked goods, and--for the kids!--asbestos cloaks. It seems a bit unlikely that a Roman emperor would send an embassy halfway around the world and tell them to pick up gifts on the way, but these are exactly the sorts of goods that merchants in the area would be carrying. Which leads to a probable conclusion that these aren't actually ambassadors, but are instead a couple merchants claiming to be ambassadors, either opportunistically or because they got picked up by a Han garrison loitering around a military post.

Beyond that there is very little in the way of information regarding individual Romans meeting individual Chinese. Another history (Liang Shu) mentions in a very offhand way that Roman merchants were common around Thailand, which is very interesting but also not very helpful given how laconic it is. There is a first century cemetery in Vagnari, southern Italy, one of the inhabitants of which was east Asian,^1 which speaks of a fascinating personal biography but not much else. A Roman merchant named Maes Tatianus reached what is probably modern Tashkent, and the Chinese general Ban Chao who led an expedition deep into central Asia dispatched the ambassador Gan Ying to find Rome--he didn't, probably turning back at the Persian Gulf. There is also the town of Liqian in Gansu, supposedly founded by Roman soldiers who were captured at the battle of Carrhae by the Parthians and eventually made their way east, where they were eventually taken by the Chinese at the battle of Zhizhi and settled in a garrison town. This story is super cool^2 but probably not true (the dates don't work out).

So unless you want to really overread the Liang Shu, there was probably not a significant presence of Romans in East Asia in any real capacity--individuals, perhaps, but not in any significant way. A great deal of Chinese silk reached Rome, and a not insignificant amount of Roman goods reached China,^3 but this doesn't seem to have led to direct, sustained contact.

A good book on this is Raoul MacLaughlin's Rome's Trade with the East, which is not without flaws but focuses more on the eastern end than any other readily available work I have seen.

^1 More accurately, his mitochondrial DNA, which is passed through the mother, has markers indicative of East Asian origin.

^2 Hollywood/HBO/Starz/Whoever, seriously, a Roman cohort having adventures in first century central Asia. Come on.

^3 Roman glass is the most notable remain found in Chinese tombs, but I haven't seen a big data analysis of how common it is.

Related: I just picked up a copy of 'The Eagle and the Dragon' which goes into this very question. Can anyone knowledgeable on the subject say how good of a history it is?

Is it wrong to call it China? Would it be like instead of saying "Roman Empire" saying "Italy"?